Big Ideas Math Book Algebra 2 Answer Key Chapter 1 Linear Functions

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Linear Functions Maintaining Mathematical Proficiency

Evaluate.

Question 1.
5 • 23 + 7

5 • 23 + 7
5.8+7
40+7
47

Question 2.
4 – 2(3 + 2)2

4 – 2(3 + 2)2
4 – 2 (25)
4  – 2 (25)
4 – 50
= – 46

Question 3.
48 ÷ 42 + $$\frac{3}{5}$$

Question 4.
50 ÷ 52 • 2

50 ÷ 52 • 2
50 ÷ 25 . 2
2 .2
4

Question 5.
$$\frac{1}{2}$$(22+ 22)

Question 6.
$$\frac{1}{6}$$(6 + 18) – 22

Graph the transformation of the figure.

Question 7.
Translate the rectangle 1 unit right and 4 units up.

Question 8.
Reflect the triangle in the y-axis. Then translate 2 units left.

Question 9.
Translate the trapezoid 3 units down. Then reflect in the x-axis.

Question 10.
ABSTRACT REASONING Give an example to show why the order of operations is important when evaluating a numerical expression. Is the order of transformations of figures important? Justify your answer.

The order of operations tells us the order to solve steps in expressions with more than one operation. First, we solve any operations inside of parentheses or brackets.

Linear Functions Maintaining Mathematical Practices

Monitoring Progress

Use a graphing calculator to graph the equation using the standard viewing window and a square viewing window. Describe any differences in the graphs.

Question 1.
y = 2x – 3

Question 2.
y = | x + 2 |

Question 3.
y = -x2 + 1

Question 4.
y = $$\sqrt{x-1}$$

Question 5.
y = x3 – 2

Question 6.
y = 0.25x3

Determine whether the viewing window is square. Explain.

Question 7.
-8 ≤ x ≤ 8, -2 ≤ y ≤ 8

Question 8.
-7 ≤ x ≤ 8, -2 ≤ y ≤ 8

Question 9.
-6 ≤ x ≤ 9, -2 ≤ y ≤ 8

Question 10.
-2 ≤ x≤ 2, -3 ≤ y ≤ 3

Question 11.
-4 ≤ x ≤ 5, -3 ≤ y ≤ 3

Question 12.
-4 ≤ x ≤ 4, -3 ≤ y ≤ 3

Lesson 1.1 Parent Functions and Transformations

Essential Question

What are the characteristics of some of the basic parent functions?
Odd. End behavior go in different directions. If a function is positive, the left side of the graph will point down and the right side will point up (increasing from left to right).

EXPLORATION 1
Identifying Basic Parent Functions
Work with a partner.
Graphs of eight basic parent functions are shown below. Classify each function as constant, linear, absolute value, quadratic, square root, cubic, reciprocal, or exponential. Justify your reasoning.

Question 2.
What are the characteristics of some of the basic parent functions?
Key common points of linear parent functions include the fact that the: Equation is y = x. Domain and range are real numbers. Slope, or rate of change, is constant.

Question 3.
Write an equation for each function whose graph is shown in Exploration 1. Then use a graphing calculator to verify that your equations are correct.

1.1 Lesson

Monitoring Progress

Question 1.
Identify the function family to which g belongs. Compare the graph of g to the graph of its parent function.

Graph the function and its parent function. Then describe the transformation.

Question 2.
g(x) = x + 3

Question 3.
h(x) = (x – 2)2

Question 4.
n(x) = – | x |

Graph the function and its parent function. Then describe the transformation.

Question 5.
g(x) = 3x

Question 6.
h(x) = $$\frac{3}{2}$$x2

Question 7.
c(x) = 0.2|x|

Use a graphing calculator to graph the function and its parent function. Then describe the transformations

Question 8.
h(x) = –$$\frac{1}{4}$$x + 5

Question 9.
d(x) = 3(x – 5)2 – 1

Question 10.
The table shows the amount of fuel in a chainsaw over time. What type of function can you use to model the data? When will the tank be empty?

Parent Functions and Transformations 1.1 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
The function f(x) = x2 is the ______ of f(x) = 2x2 – 3.

Question 2.
DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, identify the function family to which f belongs. Compare the graph of f to the graph of its parent function.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.
MODELING WITH MATHEMATICS
At 8:00 A.M., the temperature is 43°F. The temperature increases 2°F each hour for the next 7 hours. Graph the temperatures over time t (t = 0 represents 8:00 A.M.). What type of function can you use to model the data? Explain.

Question 8.
MODELING WITH MATHEMATICS
You purchase a car from a dealership for 10,000. The trade-in value of the car each year after the purchase is given by the function f(x) = 10,000 – 250x2. What type of function models the trade-in value? Answer: In Exercises 9–18, graph the function and its parent function. Then describe the transformation. Question 9. g(x) = x + 4 Answer: Question 10. f(x) = x – 6 Answer: Question 11. f(x) = x2 – 1 Answer: Question 12. h(x) = (x+ 4)2 Answer: Question 13. g(x) = | x – 5 | Answer: Question 14. f(x) = 4 + | x | Answer: Question 15. h(x) = -x2 Answer: Question 16. g(x) = -x Answer: Question 17. f(x) = 3 Answer: Question 18. f(x) = -2 Answer: In Exercises 19–26, graph the function and its parent function. Then describe the transformation. Question 19. f(x) = $$\frac{1}{3}$$x Answer: Question 20. g(x) = 4x Answer: Question 21. f(x) = 2x2 Answer: Question 22. h(x) = $$\frac{1}{3}$$x2 Answer: Question 23. h(x) = $$\frac{3}{4}$$x Answer: Question 24. g(x) = $$\frac{4}{3}$$x Answer: Question 25. h(x) = 3 | x | Answer: Question 26. f(x) = $$\frac{1}{2}$$ | x | Answer: In Exercises 27–34, use a graphing calculator to graph the function and its parent function. Then describe the transformations. Question 27. f(x) = 3x + 2 Answer: Question 28. h(x) = -x + 5 Answer: Question 29. h(x) = -3 | x | – 1 Answer: Question 30. f(x) = $$\frac{3}{4}$$ | x | + 1 Answer: Question 31. g(x) = $$\frac{1}{2}$$x2 – 6 Answer: Question 32. f(x) = 4x2 – 3 Answer: Question 33. f(x) = -(x + 3)2 + $$\frac{1}{4}$$ Answer: Question 34. g(x) = – | x – 1 | – $$\frac{1}{2}$$ Answer: ERROR ANALYSIS In Exercises 35 and 36, identify and correct the error in describing the transformation of the parent function. Question 35. Answer: Question 36. Answer: MATHEMATICAL CONNECTIONS In Exercises 37 and 38, find the coordinates of the figure after the transformation. Question 37. Translate 2 units down. Answer: Question 38. Reflect in the x-axis. Answer: USING TOOLS In Exercises 39–44, identify the function family and describe the domain and range. Use a graphing calculator to verify your answer. Question 39. g(x) = | x + 2 | – 1 Answer: Question 40. h(x) = | x – 3 | + 2 Answer: Question 41. g(x) = 3x + 4 Answer: Question 42. f(x) = -4x + 11 Answer: Question 43. f(x) = 5x2 – 2 Answer: Question 44. f(x) = -2x2 + 6 Answer: Question 45. MODELING WITH MATHEMATICS The table shows the speeds of a car as it travels through an intersection with a stop sign. What type of function can you use to model the data? Estimate the speed of the car when it is 20 yards past the intersection. Answer: Question 46. THOUGHT PROVOKING In the same coordinate plane, sketch the graph of the parent quadratic function and the graph of a quadratic function that has no x-intercepts. Describe the transformation(s) of the parent function. Answer: Question 47. USING STRUCTURE Graph the functions f(x) = | x – 4 | and g(x) = | x | – 4. Are they equivalent? Explain. Answer: Question 48. HOW DO YOU SEE IT? Consider the graphs of f, g, and h. a. Does the graph of g represent a vertical stretch or a vertical shrink of the graph of f? Explain your reasoning. b. Describe how to transform the graph of f to obtain the graph of h. Answer: Question 49. MAKING AN ARGUMENT Your friend says two different translations of the graph of the parent linear function can result in the graph of f(x) = x – 2. Is your friend correct? Explain. Answer: Question 50. DRAWING CONCLUSIONS A person swims at a constant speed of 1 meter per second. What type of function can be used to model the distance the swimmer travels? If the person has a 10-meter head start, what type of transformation does this represent? Explain. Answer: Question 51. PROBLEM SOLVING You are playing basketball with your friends. The height (in feet) of the ball above the ground t seconds after a shot is released from your hand is modeled by the function f(t) = -16t2 + 32t + 5.2. a. Without graphing, identify the type of function that models the height of the basketball. b. What is the value of t when the ball is released from your hand? Explain your reasoning. c. How many feet above the ground is the ball when it is released from your hand? Explain. Answer: Question 52. MODELING WITH MATHEMATICS The table shows the battery lives of a computer over time. What type of function can you use to model the data? Interpret the meaning of the x-intercept in this situation. Answer: Question 53. REASONING Compare each function with its parent function. State whether it contains a horizontal translation, vertical translation, both, or neither. Explain your reasoning. a. f(x) = 2 | x | – 3 b. f(x) = (x – 8)2 c. f(x) = | x + 2 | + 4 d. f(x) = 4x2 Answer: Question 54. CRITICAL THINKING Use the values -1, 0, 1, and 2 in the correct box so the graph of each function intersects the x-axis. Explain your reasoning. Answer: Maintaining Mathematical Proficiency Determine whether the ordered pair is a solution of the equation. (Skills Review Handbook) Question 55. f(x) = | x + 2 |; (1, -3) Answer: Question 56. f(x) = | x | – 3; (-2, -5) Answer: Question 57. f(x) = x – 3; (5, 2) Answer: Question 58. f(x) = x – 4; (12, 8) Answer: Find the x-intercept and the y-intercept of the graph of the equation. (Skills Review Handbook) Question 59. y = x Answer: Question 60. y = x + 2 Answer: Question 61. 3x + y = 1 Answer: Question 62. x – 2y = 8 Answer: Lesson 1.2 Transformations of Linear and Absolute Value Functions Essential Question How do the graphs of y = f(x) + k, y = f(x – h), and y = -f(x) compare to the graph of the parent function f? EXPLORATION 1 Transformations of the Parent Absolute Value Function Work with a partner. Compare the graph of the function y = | x | + k Transformation to the graph of the parent function f(x) = | x |. EXPLORATION 2 Transformations of the Parent Absolute Value Function Work with a partner. Compare the graph of the function y = | x – h | Transformation to the graph of the parent function f(x) = | x |. Parent function EXPLORATION 3 Transformation of the Parent Absolute Value Function Work with a partner. Compare the graph of the function y = – | x | Transformation to the graph of the parent function f(x) = | x | Parent function Communicate Your Answer Question 4. Transformation How do the graphs of y = f (x) + k, y = f(x – h), and y = -f(x) compare to the graph of the parent function f? Question 5. Compare the graph of each function to the graph of its parent function f. Use a graphing calculator to verify your answers are correct. a. y = $$[\sqrt{x}/latex] – 4 b. y = [\sqrt{x + 4}/latex] c. y = –[latex][\sqrt{x}/latex] d. y = x2 + 1 e. y = (x – 1)2 f. y = -x2 1.2 Lesson Monitoring Progress Write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer. Question 1. f(x) = 3x; translation 5 units up Question 2. f(x) = | x | – 3; translation 4 units to the right Question 3. f(x) = – | x + 2 | – 1; reflection in the x-axis Question 4. f(x) = [latex]\frac{1}{2}$$x+ 1; reflection in the y-axis Write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer. Question 5. f(x) = 4x+ 2; horizontal stretch by a factor of 2 Question 6. f(x) = | x | – 3; vertical shrink by a factor of $$\frac{1}{3}$$ Question 7. Let the graph of g be a translation 6 units down followed by a reflection in the x-axis of the graph of f(x) = | x |. Write a rule for g. Use a graphing calculator to check your answer. Question 8. WHAT IF? In Example 5, your revenue function is f(x) = 3x. How does this affect your profit for 100 downloads? Transformations of Linear and Absolute Value Functions 1.2 Exercises Vocabulary and Core Concept Check Question 1. COMPLETE THE SENTENCE The function g(x) = | 5x |- 4 is a horizontal ___________ of the function f(x) = | x | – 4. Answer: Question 2. WHICH ONE DOESN’T BELONG? Which transformation does not belong with the other three? Explain your reasoning. Answer: Monitoring Progress and Modeling with Mathematics In Exercises 3–8, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer. Question 3. f(x) = x – 5; translation 4 units to the left Answer: Question 4. f(x) = x + 2; translation 2 units to the right Answer: Question 5. f(x) = | 4x + 3 | + 2; translation 2 units down Answer: Question 6. f(x) = 2x – 9; translation 6 units up Answer: Question 7. f(x) = 4 – | x + 1 | Answer: Question 8. f(x) = | 4x | + 5 Answer: Question 9. WRITING Describe two different translations of the graph of f that result in the graph of g. Answer: Question 10. PROBLEM SOLVING You open a café. The function f(x) = 4000x represents your expected net income (in dollars) after being open x weeks. Before you open, you incur an extra expense of 12,000. What transformation of f is necessary to model this situation? How many weeks will it take to pay off the extra expense? Answer: In Exercises 11–16, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer. Question 11. f(x) = -5x+ 2; reflection in the x-axis Answer: Question 12. f(x) = $$\frac{1}{2}$$x – 3; reflection in the x-axis Answer: Question 13. f(x) = | 6x | – 2; reflection in the y-axis Answer: Question 14. f(x) = | 2x – 1 | + 3; reflection in the y-axis Answer: Question 15. f(x) = -3 + | x – 11 |; reflection in the y-axis Answer: Question 16. f(x) = -x+ 1; reflection in the y-axis In Exercises 17–22, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer. Question 17. f(x) = x + 2; vertical stretch by a factor of 5 Answer: Question 18. f(x) = 2x+ 6; vertical shrink by a factor of $$\frac{1}{2}$$ Answer: Question 19. f(x) = | 2x | + 4; horizontal shrink by a factor of $$\frac{1}{2}$$ Answer: Question 20. f(x) = | x+ 3 | ; horizontal stretch by a factor of 4 Answer: Question 21. f(x) = -2 | x – 4 | + 2 Answer: Question 22. f(x) = 6 – x Answer: ANALYZING RELATIONSHIPS In Exercises 23–26, match the graph of the transformation of f with the correct equation shown. Explain your reasoning. Question 23. Answer: Question 24. Answer: Question 25. Answer: Question 26. Answer: A. y = 2f(x) B. y = f(2x) C. y = f(x + 2) D. y = f(x) + 2 In Exercises 27–32, write a function g whose graph represents the indicated transformations of the graph of f. Question 27. f(x) = x; vertical stretch by a factor of 2 followed by a translation 1 unit up Answer: Question 28. f(x) = x; translation 3 units down followed by a vertical shrink by a factor of $$\frac{1}{3}$$ Answer: Question 29. f(x) = | x | ; translation 2 units to the right followed by a horizontal stretch by a factor of 2 Answer: Question 30. f(x) = | x |; reflection in the y-axis followed by a translation 3 units to the right Answer: Question 31. f(x) = | x | Answer: Question 32. f(x) = | x | Answer: ERROR ANALYSIS In Exercises 33 and 34, identify and correct the error in writing the function g whose graph represents the indicated transformations of the graph of f. Question 33. Answer: Question 34. Answer: Question 35. MAKING AN ARGUMENT Your friend claims that when writing a function whose graph represents a combination of transformations, the order is not important. Is your friend correct? Justify your answer. Answer: Question 36. MODELING WITH MATHEMATICS During a recent period of time, bookstore sales have been declining. The sales (in billions of dollars) can be modeled by the function f(t) = –$$\frac{7}{5}$$t + 17.2, where t is the number of years since 2006. Suppose sales decreased at twice the rate. How can you transform the graph of f to model the sales? Explain how the sales in 2010 are affected by this change. Answer: MATHEMATICAL CONNECTIONS For Exercises 37–40, describe the transformation of the graph of f to the graph of g. Then find the area of the shaded triangle. Question 37. f(x) = | x – 3 | Answer: Question 38. f(x) = – | x | – 2 Answer: Question 39. f(x) = -x + 4 Answer: Question 40. f(x) = x – 5 Answer: Question 41. ABSTRACT REASONING The functions f(x) = mx + b and g(x) = mx + c represent two parallel lines. a. Write an expression for the vertical translation of the graph of f to the graph of g. b. Use the definition of slope to write an expression for the horizontal translation of the graph of f to the graph of g. Answer: Question 42. HOW DO YOU SEE IT? Consider the graph of f(x) = mx + b. Describe the effect each transformation has on the slope of the line and the intercepts of the graph. a. Reflect the graph of f in the y-axis. b. Shrink the graph of f vertically by a factor of $$\frac{1}{3}$$. c. Stretch the graph of f horizontally by a factor of 2. Answer: Question 43. REASONING The graph of g(x) = -4 |x | + 2 is a reflection in the x-axis, vertical stretch by a factor of 4, and a translation 2 units down of the graph of its parent function. Choose the correct order for the transformations of the graph of the parent function to obtain the graph of g. Explain your reasoning. Answer: Question 44. THOUGHT PROVOKING You are planning a cross-country bicycle trip of 4320 miles. Your distance d (in miles) from the halfway point can be modeled by d = 72 |x – 30 |, where x is the time (in days) and x = 0 represents June 1. Your plans are altered so that the model is now a right shift of the original model. Give an example of how this can happen. Sketch both the original model and the shifted model. Answer: Question 45. CRITICAL THINKING Use the correct value 0, -2, or 1 with a, b, and c so the graph of g(x) = a|x – b | + c is a reflection in the x-axis followed by a translation one unit to the left and one unit up of the graph of f(x) = 2 |x – 2 | + 1. Explain your reasoning. Answer: Maintaining Mathematical Proficiency Evaluate the function for the given value of x. (Skills Review Handbook) Question 46. f(x) = x + 4; x = 3 Answer: Question 47. f(x) = 4x – 1; x = -1 Answer: Question 48. f(x) = -x + 3; x = 5 Answer: Question 49. f(x) = -2x – 2; x = -1 Answer: Create a scatter plot of the data. (Skills Review Handbook) Question 50. Answer: Question 51. Answer: Linear Functions Study Skills Taking Control of Your Class Time 1.1 – 1.2 What Did You Learn? Core Vocabulary Core Concepts Section 1.1 Section 1.2 Mathematical Practices Question 1. How can you analyze the values given in the table in Exercise 45 on page 9 to help you determine what type of function models the data? Question 2. Explain how you would round your answer in Exercise 10 on page 16 if the extra expense is 13,500. Study Skills Taking Control of Your Class Time Question 1. Sit where you can easily see and hear the teacher, and the teacher can see you. Question 2. Pay attention to what the teacher says about math, not just what is written on the board. Question 3. Ask a question if the teacher is moving through the material too fast. Question 4. Try to memorize new information while learning it. Question 5. Ask for clarification if you do not understand something. Question 6. Think as intensely as if you were going to take a quiz on the material at the end of class. Question 7. Volunteer when the teacher asks for someone to go up to the board. Question 8. At the end of class, identify concepts or problems for which you still need clarification. Question 9. Use the tutorials at BigIdeasMath.com for additional help. Linear Functions 1.1-1.2 Quiz Identify the function family to which g belongs. Compare the graph of the function to the graph of its parent function. (Section 1.1) Question 1. Question 2. Question 3. Graph the function and its parent function. Then describe the transformation. (Section 1.1) Question 4. f(x) = $$\frac{3}{2}$$ Question 5. f(x) = 3x Question 6. f(x) = 2(x – 1)2 Question 7. f(x) = – | x + 2 | – 7 Question 8. f(x) = $$\frac{1}{4}$$x2 + 1 Question 9. f(x) = –$$\frac{1}{2}$$x – 4 Write a function g whose graph represents the indicated transformation of the graph of f. (Section 1.2) Question 10. f(x) = 2x + 1; translation 3 units up Question 11. f(x) = -3 | x – 4 | ; vertical shrink by a factor of $$\frac{1}{2}$$ Question 12. f(x) = 3 | x + 5 |; reflection in the x-axis Question 13. f(x) = $$\frac{1}{3}$$x – $$\frac{2}{3}$$ ; translation 4 units left Write a function g whose graph represents the indicated transformations of the graph of f. (Section 1.2) Question 14. Let g be a translation 2 units down and a horizontal shrink by a factor of $$\frac{2}{3}$$ of the graph of f(x) =x. Question 15. Let g be a translation 9 units down followed by a reflection in the y-axis of the graph of f(x) = x. Question 16. Let g be a reflection in the x-axis and a vertical stretch by a factor of 4 followed by a translation 7 units down and 1 unit right of the graph of f(x) = | x |. Question 17. Let g be a translation 1 unit down and 2 units left followed by a vertical shrink by a factor of $$\frac{1}{2}$$ of the graph of f(x) = | x |. Question 18. The table shows the total distance a new car travels each month after it is purchased. What type of function can you use to model the data? Estimate the mileage after 1 year. (Section 1.1) Question 19. The total cost of an annual pass plus camping for x days in a National Park can be modeled by the function f(x) = 20x+ 80. Senior citizens pay half of this price and receive an additional 30 discount. Describe how to transform the graph of f to model the total cost for a senior citizen. What is the total cost for a senior citizen to go camping for three days? (Section 1.2) Lesson 1.3 Modeling with Linear Functions Essential Question How can you use a linear function to model and analyze a real-life situation? EXPLORATION 1 Modeling with a Linear Function Work with a partner. A company purchases a copier for 12,000. The spreadsheet shows how the copier depreciates over an 8-year period. a. Write a linear function to represent the value V of the copier as a function of the number t of years. b. Sketch a graph of the function. Explain why this type of depreciation is called straight line depreciation. c. Interpret the slope of the graph in the context of the problem. EXPLORATION 2 Modeling with Linear Functions Work with a partner. Match each description of the situation with its corresponding graph. Explain your reasoning. a. A person gives 20 per week to a friend to repay a 200 loan. b. An employee receives 12.50 per hour plus 2 for each unit produced per hour. c. A sales representative receives 30 per day for food plus 0.565 for each mile driven. d. A computer that was purchased for 750 depreciates 100 per year. Communicate Your Answer Question 3. How can you use a linear function to model and analyze a real-life situation? Question 4. Use the Internet or some other reference to find a real-life example of straight line depreciation. a. Use a spreadsheet to show the depreciation. b. Write a function that models the depreciation. c. Sketch a graph of the function. 1.3 Lesson Monitoring Progress Question 1. The graph shows the remaining balance y on a car loan after making x monthly payments. Write an equation of the line and interpret the slope and y-intercept. What is the remaining balance after 36 payments? Question 2. WHAT IF? Maple Ridge charges a rental fee plus a 10 fee per student. The total cost is 1900 for 140 students. Describe the number of students that must attend for the total cost at Maple Ridge to be less than the total costs at the other two venues. Use a graph to justify your answer. Question 3. The table shows the humerus lengths (in centimeters) and heights (in centimeters) of several females. a. Do the data show a linear relationship? If so, write an equation of a line of fit and use it to estimate the height of a female whose humerus is 40 centimeters long. b. Use the linear regression feature on a graphing calculator to find an equation of the line of best fit for the data. Estimate the height of a female whose humerus is 40 centimeters long. Compare this height to your estimate in part (a). Modeling with Linear Functions 1.3 Exercises Question 1. COMPLETE THE SENTENCE The linear equation y = $$\frac{1}{2}$$x + 3 is written in ____________ form. Answer: Question 2. VOCABULARY A line of best fit has a correlation coefficient of -0.98. What can you conclude about the slope of the line? Answer: Monitoring Progress and Modeling with Mathematics In Exercises 3–8, use the graph to write an equation of the line and interpret the slope. Question 3. Answer: Question 4. Answer: Question 5. Answer: Question 6. Answer: Question 7. Answer: Question 8. Answer: Question 9. MODELING WITH MATHEMATICS Two newspapers charge a fee for placing an advertisement in their paper plus a fee based on the number of lines in the advertisement. The table shows the total costs for different length advertisements at the Daily Times. The total cost y (in dollars) for an advertisement that is x lines long at the Greenville Journal is represented by the equation y = 2x + 20. Which newspaper charges less per line? How many lines must be in an advertisement for the total costs to be the same? Answer: Question 10. PROBLEM SOLVING While on vacation in Canada, you notice that temperatures are reported in degrees Celsius. You know there is a linear relationship between Fahrenheit and Celsius, but you forget the formula. From science class, you remember the freezing point of water is 0°C or 32°F, and its boiling point is 100°C or 212°F. a. Write an equation that represents degrees Fahrenheit in terms of degrees Celsius. b. The temperature outside is 22°C. What is this temperature in degrees Fahrenheit? c. Rewrite your equation in part (a) to represent degrees Celsius in terms of degrees Fahrenheit. d. The temperature of the hotel pool water is 83°F. What is this temperature in degrees Celsius? Answer: ERROR ANALYSIS In Exercises 11 and 12, describe and correct the error in interpreting the slope in the context of the situation. Question 11. Answer: Question 12. Answer: In Exercises 13–16, determine whether the data show a linear relationship. If so, write an equation of a line of fit. Estimate y when x = 15 and explain its meaning in the context of the situation. Question 13. Answer: Question 14. Answer: Question 15. Answer: Question 16. Answer: Question 17. MODELING WITH MATHEMATICS The data pairs (x, y) represent the average annual tuition y (in dollars) for public colleges in the United States x years after 2005. Use the linear regression feature on a graphing calculator to find an equation of the line of best fit. Estimate the average annual tuition in 2020. Interpret the slope and y-intercept in this situation. Answer: Question 18. MODELING WITH MATHEMATICS The table shows the numbers of tickets sold for a concert when different prices are charged. Write an equation of a line of fit for the data. Does it seem reasonable to use your model to predict the number of tickets sold when the ticket price is 85? Explain. Answer: USING TOOLS In Exercises 19–24, use the linear regression feature on a graphing calculator to find an equation of the line of best fit for the data. Find and interpret the correlation coefficient. Question 19. Answer: Question 20. Answer: Question 21. Answer: Question 22. Answer: Question 23. Answer: Question 24. Answer: Question 25. OPEN-ENDED Give two real-life quantities that have (a) a positive correlation, (b) a negative correlation, and (c) approximately no correlation. Explain. Answer: Question 26. HOW DO YOU SEE IT? You secure an interest-free loan to purchase a boat. You agree to make equal monthly payments for the next two years. The graph shows the amount of money you still owe. a. What is the slope of the line? What does the slope represent? b. What is the domain and range of the function? What does each represent? c. How much do you still owe after making payments for 12 months? Answer: Question 27. MAKING AN ARGUMENT A set of data pairs has a correlation coefficient r = 0.3. Your friend says that because the correlation coefficient is positive, it is logical to use the line of best fit to make predictions. Is your friend correct? Explain your reasoning. Answer: Question 28. THOUGHT PROVOKING Points A and B lie on the line y = -x + 4. Choose coordinates for points A, B, and C where point C is the same distance from point A as it is from point B. Write equations for the lines connecting points A and C and points B and C. Answer: Question 29. ABSTRACT REASONING If x and y have a positive correlation, and y and z have a negative correlation, then what can you conclude about the correlation between x and z? Explain. Answer: Question 30. MATHEMATICAL CONNECTIONS Which equation has a graph that is a line passing through the point (8, -5) and is perpendicular to the graph of y = -4x + 1? A. y = $$\frac{1}{4}$$x – 5 B. y = -4x + 27 C. y = –$$\frac{1}{4}$$x – 7 D. y = $$\frac{1}{4}$$x – 7 Answer: Question 31. PROBLEM SOLVING You are participating in an orienteering competition. The diagram shows the position of a river that cuts through the woods. You are currently 2 miles east and 1 mile north of your starting point, the origin. What is the shortest distance you must travel to reach the river? Answer: Question 32. ANALYZING RELATIONSHIPS Data from North American countries show a positive correlation between the number of personal computers per capita and the average life expectancy in the country. a. Does a positive correlation make sense in this situation? Explain. b. Is it reasonable to conclude that giving residents of a country personal computers will lengthen their lives? Explain. Answer: Maintaining Mathematical Proficiency Solve the system of linear equations in two variables by elimination or substitution. (Skills Review Handbook) Question 33. 3x + y = 7 -2x – y = 9 Answer: Question 34. 4x + 3y = 2 2x – 3y = 1 Answer: Question 35. 2x + 2y = 3 x = 4y – 1 Answer: Question 36. y = 1 + x 2x + y = -2 Answer: Question 37. $$\frac{1}{2}$$x + 4y = 4 2x – y = 1 Answer: Question 38. y = x – 4 4x + y = 26 Answer: Lesson 1.4 Solving Linear Systems Essential Question How can you determine the number of solutions of a linear system? A linear system is consistent when it has at least one solution. A linear system is inconsistent when it has no solution. EXPLORATION 1 Recognizing Graphs of Linear Systems Work with a partner. Match each linear system with its corresponding graph. Explain your reasoning. Then classify the system as consistent or inconsistent. a. 2x – 3y = 3 -4x + 6y = 6 b. 2x – 3y = 3 x + 2y = 5 c. 2x – 3y = 3 -4x + 6y = 6 EXPLORATION 2 Solving Systems of Linear Equations Work with a partner. Solve each linear system by substitution or elimination. Then use the graph of the system below to check your solution. a. 2x + y = 5 x – y = 1 b. x+ 3y = 1 -x + 2y = 4 c. x + y = 0 3x + 2y = 1 Communicate Your Answer Question 3. How can you determine the number of solutions of a linear system? Question 4. Suppose you were given a system of three linear equations in three variables. Explain how you would approach solving such a system. Question 5. Apply your strategy in Question 4 to solve the linear system. 1.4 Lesson Monitoring Progress Question 1. x – 2y + z = -11 3x + 2y – z = 7 -x + 2y + 4z = -9 Question 2. x + y – z = -1 4x + 4y – 4z = -2 3x + 2y + z = 0 Question 3. x + y + z = 8 x – y + z = 8 2x + y + 2z = 16 Question 4. In Example 3, describe the solutions of the system using an ordered triple in terms of y. Question 5. WHAT IF? On the first day, 10,000 tickets sold, generating 356,000 in revenue. The number of seats sold in Sections A and B are the same. How many lawn seats are still available? Solving Linear Systems 1.4 Exercises Vocabulary and Core Concept Check Question 1. VOCABULARY The solution of a system of three linear equations is expressed as a(n)__________. Answer: Question 2. WRITING Explain how you know when a linear system in three variables has infinitely many solutions. Answer: Monitoring Progress and Modeling with Mathematics In Exercises 3–8, solve the system using the elimination method. Question 3. x + y – 2z = 5 -x + 2y + z = 2 2x + 3y – z = 9 Answer: Question 4. x + 4y – 6z = -1 2x – y + 2z = -7 -x + 2y – 4z = 5 Answer: Question 5. 2x + y – z = 9 -x + 6y + 2z = -17 5x + 7y + z = 4 Answer: Question 6. 3x + 2y – z = 8 -3x + 4y + 5z = -14 x – 3y + 4z = -14 Answer: Question 7. 2x + 2y + 5z = -1 2x – y + z = 2 2x + 4y – 3z = 14 Answer: Question 8. 3x + 2y – 3z = -2 7x – 2y + 5z = -14 2x + 4y + z = 6 Answer: ERROR ANALYSIS In Exercises 9 and 10, describe and correct the error in the first step of solving the system of linear equations. Question 9. Answer: Question 10. Answer: In Exercises 11–16, solve the system using the elimination method. Question 11. 3x – y + 2z = 4 6x – 2y + 4z = -8 2x – y + 3z = 10 Answer: Question 12. 5x + y – z = 6 x + y + z = 2 12x + 4y = 10 Answer: Question 13. x + 3y – z = 2 x + y – z = 0 3x + 2y – 3z = -1 Answer: Question 14. x + 2y – z = 3 -2x – y + z = -1 6x – 3y – z = -7 Answer: Question 15. x + 2y + 3z = 4 -3x + 2y – z = 12 -2x – 2y – 4z = -14 Answer: Question 16. -2x – 3y + z = -6 x + y – z = 5 7x + 8y – 6z = 31 Answer: Question 17. MODELING WITH MATHEMATICS Three orders are placed at a pizza shop. Two small pizzas, a liter of soda, and a salad cost 14; one small pizza, a liter of soda, and three salads cost 15; and three small pizzas, a liter of soda, and two salads cost 22. How much does each item cost? Answer: Question 18. MODELING WITH MATHEMATICS Sam’s Furniture Store places the following advertisement in the local newspaper. Write a system of equations for the three combinations of furniture. What is the price of each piece of furniture? Explain. Answer: In Exercises 19–28, solve the system of linear equations using the substitution method. Question 19. -2x + y + 6z = 1 3x + 2y + 5z = 16 7x + 3y – 4z = 11 Answer: Question 20. x – 6y – 2z = -8 -x + 5y + 3z = 2 3x – 2y – 4z = 18 Answer: Question 21. x + y + z = 4 5x + 5y + 5z = 12 x – 4y + z = 9 Answer: Question 22. x + 2y = -1 -x + 3y + 2z = -4 -x + y – 4z = 10 Answer: Question 23. 2x – 3y + z = 10 y + 2z = 13 z = 5 Answer: Question 24. x = 4 x + y = -6 4x – 3y + 2z = 26 Answer: Question 25. x + y – z = 4 3x + 2y + 4z = 17 -x + 5y + z = 8 Answer: Question 26. 2x – y – z = 15 4x + 5y + 2z = 10 -x – 4y + 3z = -20 Answer: Question 27. 4x + y + 5z = 5 8x + 2y + 10z = 10 x – y – 2z = -2 Answer: Question 28. x + 2y – z = 3 2x + 4y – 2z = 6 -x – 2y + z = -6 Answer: Question 29. PROBLEM SOLVING The number of left-handed people in the world is one-tenth the number of right-handed people. The percent of right-handed people is nine times the percent of left-handed people and ambidextrous people combined. What percent of people are ambidextrous? Answer: Question 30. MODELING WITH MATHEMATICS Use a system of linear equations to model the data in the following newspaper article. Solve the system to find how many athletes finished in each place. Answer: Question 31. WRITING Explain when it might be more convenient to use the elimination method than the substitution method to solve a linear system. Give an example to support your claim. Answer: Question 32. REPEATED REASONING Using what you know about solving linear systems in two and three variables, plan a strategy for how you would solve a system that has four linear equations in four variables. Answer: MATHEMATICAL CONNECTIONS In Exercises 33 and 34, write and use a linear system to answer the question. Question 33. The triangle has a perimeter of 65 feet. What are the lengths of sides ℓ, m, and n? Answer: Question 34. What are the measures of angles A, B, and C? Answer: Question 35. OPEN-ENDED Consider the system of linear equations below. Choose nonzero values for a, b, and c so the system satisfies the given condition. Explain your reasoning. x + y + z = 2 ax + by + cz = 10 x – 2y + z = 4 a. The system has no solution. b. The system has exactly one solution. c. The system has infinitely many solutions. Answer: Question 36. MAKING AN ARGUMENT A linear system in three variables has no solution. Your friend concludes that it is not possible for two of the three equations to have any points in common. Is your friend correct? Explain your reasoning. Answer: Question 37. PROBLEM SOLVING A contractor is hired to build an apartment complex. Each 840-square-foot unit has a bedroom, kitchen, and bathroom. The bedroom will be the same size as the kitchen. The owner orders 980 square feet of tile to completely cover the floors of two kitchens and two bathrooms. Determine how many square feet of carpet is needed for each bedroom. Answer: Question 38. THOUGHT PROVOKING Does the system of linear equations have more than one solution? Justify your answer. 4x + y + z = 0 2x + $$\frac{1}{2}$$y – 3z = 0 -x – $$\frac{1}{4}$$y – z = 0 Answer: Question 39. PROBLEM SOLVING A florist must make 5 identical bridesmaid bouquets for a wedding. The budget is 160, and each bouquet must have 12 flowers. Roses cost 2.50 each, lilies cost 4 each, and irises cost 2 each. The florist wants twice as many roses as the other two types of flowers combined. a. Write a system of equations to represent this situation, assuming the florist plans to use the maximum budget. b. Solve the system to find how many of each type of flower should be in each bouquet. c. Suppose there is no limitation on the total cost of the bouquets. Does the problem still have exactly one solution? If so, find the solution. If not, give three possible solutions. Answer: Question 40. HOW DO YOU SEE IT? Determine whether the system of equations that represents the circles has no solution, one solution, or infinitely many solutions. Explain your reasoning. Answer: Question 41. CRITICAL THINKING Find the values of a, b, and c so that the linear system shown has (-1, 2, -3) as its only solution. Explain your reasoning. x + 2y – 3z = a – x – y + z = b 2x + 3y – 2z = c Answer: Question 42. ANALYZING RELATIONSHIPS Determine which arrangement(s) of the integers -5, 2, and 3 produce a solution of the linear system that consist of only integers. Justify your answer. x – 3y + 6z = 21 _x + _y + _z = -30 2x – 5y + 2z = -6 Answer: Question 43. ABSTRACT REASONING Write a linear system to represent the first three pictures below. Use the system to determine how many tangerines are required to balance the apple in the fourth picture. Note:The first picture shows that one tangerine and one apple balance one grapefruit. Answer: Maintaining Mathematical Proficiency Simplify. (Skills Review Handbook) Question 44. (x – 2)2 Answer: Question 45. (3m + 1)2 Answer: Question 46. (2z – 5)2 Answer: Question 47. (4 – y)2 Answer: Write a function g described by the given transformation of f(x) =∣x∣− 5.(Section 1.2) Question 48. translation 2 units to the left Answer: Question 49. reflection in the x-axis Answer: Question 50. translation 4 units up Answer: Question 51. vertical stretch by a factor of 3 Answer: Linear Functions Performance Task: Secret of the Hanging Baskets 1.3–1.4 What Did You Learn? Core Vocabulary Core Concepts Section 1.3 Writing an Equation of a Line, p. 22 Finding a Line of Fit, p. 24 Section 1.4 Solving a Three-Variable System, p. 31 Solving Real-Life Problems, p. 33 Mathematical Practices Question 1. Describe how you can write the equation of the line in Exercise 7 on page 26 using only one of the labeled points. Question 2. How did you use the information in the newspaper article in Exercise 30 on page 35 to write a system of three linear equations? Question 3. Explain the strategy you used to choose the values for a, b, and c in Exercise 35 part (a) on page 35. Performance Task Secret of the Hanging Baskets A carnival game uses two baskets hanging from springs at different heights. Next to the higher basket is a pile of baseballs. Next to the lower basket is a pile of golf balls. The object of the game is to add the same number of balls to each basket so that the baskets have the same height. But there is a catch—you only get one chance. What is the secret to winning the game? To explore the answers to this question and more, go to BigIdeasMath.com. Linear Functions Chapter Review Graph the function and its parent function. Then describe the transformation. Question 1. f(x) = x + 3 Question 2. g(x) = | x | – 1 Question 3. h(x) = $$\frac{1}{2}$$x2 Question 4. h(x) = 4 Question 5. f(x) = -| x | – 3 Question 6. g(x) = -3(x + 3)2 Write a function g whose graph represents the indicated transformations of the graph of f. Use a graphing calculator to check your answer. Question 7. f(x) = | x |; reflection in the x-axis followed by a translation 4 units to the left Question 8. f(x) = | x | ; vertical shrink by a factor of $$\frac{1}{2}$$ followed by a translation 2 units up Question 9. f(x) = x; translation 3 units down followed by a reflection in the y-axis Question 10. The table shows the total number y (in billions) of U.S. movie admissions each year for x years. Use a graphing calculator to find an equation of the line of best fit for the data. Question 11. You ride your bike and measure how far you travel. After 10 minutes, you travel 3.5 miles. After 30 minutes, you travel 10.5 miles. Write an equation to model your distance. How far can you ride your bike in 45 minutes? Question 12. x + y + z = 3 -x + 3y + 2z = -8 x = 4z Question 13. 2x – 5y – z = 17 x + y + 3z = 19 -4x + 6y + z = -20 Question 14. x + y + z = 2 2x – 3y + z = 11 -3x + 2y – 2z = -13 Question 15. x + 4y – 2z = 3 x + 3y + 7z = 1 2x + 9y – 13z = 2 Question 16. x – y + 3z = 6 x – 2y = 5 2x – 2y + 5z = 9 Question 17. x + 2y = 4 x + y + z = 6 3x + 3y + 4z = 28 Question 18. A school band performs a spring concert for a crowd of 600 people. The revenue for the concert is 3150. There are 150 more adults at the concert than students. How many of each type of ticket are sold? Linear Functions Chapter Test Write an equation of the line and interpret the slope and y-intercept. Question 1. Question 2. Solve the system. Check your solution, if possible. Question 3. -2x + y + 4z = 5 x + 3y – z = 2 4x + y – 6z = 11 Question 4. y = $$\frac{1}{2}$$z x + 2y + 5z = 2 3x + 6y – 3z = 9 Question 5. x – y + 5z = 3 2x + 3y – z = 2 -4x – y – 9z = -8 Graph the function and its parent function. Then describe the transformation. Question 6. Question 8. f(x) = 4 Match the transformation of f(x) = x with its graph. Then write a rule for g. Question 9. g(x) = 2f(x) + 3 Question 10. g(x) = 3f(x) – 2 Question 11. g(x) = -2f(x) – 3 Question 12. A bakery sells doughnuts, muffins, and bagels. The bakery makes three times as many doughnuts as bagels. The bakery earns a total of 150 when all 130 baked items in stock are sold. How many of each item are in stock? Justify your answer. Question 13. A fountain with a depth of 5 feet is drained and then refilled. The water level (in feet) after t minutes can be modeled by f(t) = $$\frac{1}{4}$$|t – 20 |. A second fountain with the same depth is drained and filled twice as quickly as the first fountain. Describe how to transform the graph of f to model the water level in the second fountain after t minutes. Find the depth of each fountain after 4 minutes. Justify your answers. Linear Functions Cumulative Assessment Question 1. Describe the transformation of the graph of f(x) = 2x – 4 represented in each graph. Question 2. The table shows the tuition costs for a private school between the years 2010 and 2013. a. Verify that the data show a linear relationship. Then write an equation of a line of fit. b. Interpret the slope and y-intercept in this situation. c. Predict the cost of tuition in 2015. Question 3. Your friend claims the line of best fit for the data shown in the scatter plot has a correlation coefficient close to 1. Is your friend correct? Explain your reasoning. Question 4. Order the following linear systems from least to greatest according to the number of solutions. A. 2x + 4y – z = 7 14x + 28y – 7z = 49 -x + 6y + 12z = 13 B. 3x – 3y + 3z = 5 -x + y – z = 5 -x + y – z = 8 14x – 3y + 12z = 108 C. 4x – y + 2z = 18 -x + 2y + z = 11 3x + 3y – 4z = 44 Question 5. You make a DVD of three types of shows: comedy, drama, and reality-based. An episode of a comedy lasts 30 minutes, while a drama and a reality-based episode each last 60 minutes. The DVDs can hold 360 minutes of programming. a. You completely fill a DVD with seven episodes and include twice as many episodes of a drama as a comedy. Create a system of equations that models the situation. b. How many episodes of each type of show are on the DVD in part (a)? c. You completely fill a second DVD with only six episodes. Do the two DVDs have a different number of comedies? dramas? reality-based episodes? Explain. Question 6. The graph shows the height of a hang glider over time. Which equation models the situation? A. y + 450 = 10x B. 10y = -x+ 450 C. $$\frac{1}{10}$$y = -x + 450 D. 10x + y = 450 Question 7. Let f(x) = x and g(x) = -3x – 4. Select the possible transformations (in order) of the graph of f represented by the function g. A. reflection in the x-axis B. reflection in the y-axis C. vertical translation 4 units down D. horizontal translation 4 units right E. horizontal shrink by a factor of $$\frac{1}{3}$$ F. vertical stretch by a factor of 3 Question 8. Choose the correct equality or inequality symbol which completes the statement below about the linear functions f and g. Explain your reasoning. Big Ideas Math Algebra 2 Answers Chapter 3 Quadratic Equations and Complex Numbers Have you worried about the practice sessions and scores of the upcoming exams? Then, to eliminate your worries and show you the success path here comes the perfect study guide ie., Big Ideas Math Algebra 2 Answers Chapter 3 Quadratic Equations and Complex Numbers. Take the homework help from this BIM Algebra 2 Answers materials and secure the highest marks in the exams. Ch 3 Quadratic Equations and Complex Numbers Big Ideas Math Textbook Algebra 2 Answer Key cover topic-wise exercise questions, tests, review, a performance task, quiz, assessments, etc. You can learn and gain more subject knowledge with the help of BIM Book Algebra 2 Answer Key Chapter 3 Quadratic Equations and Complex Numbers. So, check out the below modules and study well. Big Ideas Math Book Algebra 2 Answer Key Chapter 3 Quadratic Equations and Complex Numbers Topic-wise Big Ideas Math Textbook Algebra 2 Ch 3 Quadratic Equations and Complex Numbers Solution Key is listed here in the form of links. Click on the links and prepare well for the exams with the help of subject experts provided BIM Algebra 2 Solutions. This Big Ideas Math Book Algebra 2 Ch 3 Answers Key is prepared as per the guidelines of the Common core curriculum and explained in a simple manner. So, have a glance at these exercise questions and get an in-depth knowledge of the subject & perform all the exam with full confidence. Quadratic Equations and Complex Numbers Maintaining Mathematical Proficiency Simplify the expression. Question 1. $$\sqrt{27}$$ Answer: Question 2. –$$\sqrt{112}$$ Answer: Question 3. $$\sqrt{\frac{11}{64}}$$ Answer: Question 4. $$\sqrt{\frac{147}{100}}$$ Answer: Question 5. $$\sqrt{\frac{18}{49}}$$ Answer: Question 6. –$$\sqrt{\frac{65}{121}}$$ Answer: Question 7. $$\sqrt{80}$$ Answer: Question 8. $$\sqrt{32}$$ Answer: Factor the polynomial. Question 9. x2 − 36 Answer: X2 – 36 x2 – 62 (x +6) (x- 6) Question 10. x2 − 9 Answer: x2 – 9 x2 – 32 (x-3) ( x+3) Question 11. 4x2 − 25 Answer: 4x2 – 25 2x2 – 5 (2x +5) ( 2x – 5) Question 12. x2 − 22x + 121 Answer: X2– 22x +121 x2 – 11x -11x +121 x(x-11)-11(x-11) (x-11)(x-11) Question 13. x2 + 28x + 196 Answer: X2 + 28x +196x x2+14x+14x+196 x(x+14)+14(x+14) (x+14)(x+14) Question 14. 49x2 + 210x + 225 Answer: Question 15. ABSTRACT REASONING Determine the possible integer values of a and c for which the trinomial ax2+ 8x+c is factorable using the Perfect Square Trinomial Pattern. Explain your reasoning. Answer: The term ‘a’ is referred to as the leading coefficient, while ‘c‘ is the absolute term of f (x). Every quadratic equation has two values of the unknown variable, Quadratic Equations and Complex Numbers Mathematical Practices Mathematically proficient students recognize the limitations of technology Monitoring Progress Question 1. Explain why the second viewing window in Example 1 shows gaps between the upper and lower semicircles, but the third viewing window does not show gaps. Answer: Use a graphing calculator to draw an accurate graph of the equation. Explain your choice of viewing window. Question 2. y = $$\sqrt{x^{2}-1.5}$$ Answer: Question 3. y = $$\sqrt{x-2.5}$$ Answer: Question 4. x2 + y2= 12.25 Answer: Question 5. x2 + y2 = 20.25 Answer: Question 6. x2 + 4y2 = 12.25 Answer: Question 7. 4x2 + y2 = 20.25 Answer: Lesson 3.1 Solving Quadratic Equations Essential Question How can you use the graph of a quadratic equation to determine the number of real solutions of the equation? EXPLORATION 1 Matching a Quadratic Function with Its Graph Work with a partner. Match each quadratic function with its graph. Explain your reasoning. Determine the number of x-intercepts of the graph. a. f(x) = x2 − 2x b. f(x) = x2 − 2x + 1 c. f(x) = x2 − 2x + 2 d. f(x) = −x2 + 2x e. f(x) = −x2 + 2x − 1 f. f(x) = −x2 + 2x − 2 EXPLORATION 2 Solving Quadratic Equations Work with a partner. Use the results of Exploration 1 to find the real solutions (if any) of each quadratic equation. a. x2 − 2x = 0 b. x2 − 2x + 1 = 0 c. x2 − 2x + 2 = 0 d. −x2 + 2x = 0 e. −x2 + 2x − 1 = 0 f. −x2 + 2x − 2 = 0 Communicate Your Answer Question 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation? Answer: Question 4. How many real solutions does the quadratic equation x2 + 3x + 2 = 0 have? How do you know? What are the solutions? Answer: Parabolas have a highest or a lowest point called the Vertex Monitoring Progress Solve the equation by graphing. Question 1. x2 − 8x + 12 = 0 Answer: Question 2. 4x2 − 12x + 9 = 0 Answer: Question 3. $$\frac{1}{2}$$x2 = 6x − 20 Answer: Solve the equation using square roots. Question 4. $$\frac{2}{3}$$x2 + 14 = 20 Answer: Question 5. −2x2 + 1 = −6 Answer: Question 6. 2(x − 4)2 = −5 Answer: Solve the equation by factoring. Question 7. x2 + 12x + 35 = 0 Answer: Question 8. 3x2 − 5x = 2 Answer: Find the zero(s) of the function. Question 9. f(x) = x2 − 8x Answer: Question 10. f(x) = 4x2 + 28x + 49 Answer: Question 11. WHAT IF? The magazine initially charges 21 per annual subscription. How much should the magazine charge to maximize annual revenue? What is the maximum annual revenue? Answer: Question 12. WHAT IF? The egg container is dropped from a height of 80 feet. How does this change your answers in parts (a) and (b)? Answer: Solving Quadratic Equations 3.1 Exercises Vocabulary and Core Concept Check Question 1. WRITING Explain how to use graphing to find the roots of the equation ax2 + bx + c = 0. Answer: Question 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. Monitoring Progress and Modeling with Mathematics In Exercises 3–12, solve the equation by graphing. Question 3. x2 + 3x + 2 = 0 Answer: Question 4. −x2 + 2x + 3 = 0 Answer: Question 5. y = x2 − 9 Answer: Question 6. −8 = −x2 − 4 Answer: Question 7. 8x = −4 − 4x2 Answer: Question 8. 3x2 = 6x − 3 Answer: Question 9. 7 = −x2 − 4x Answer: Question 10. 2x = x2 + 2 Answer: Question 11. $$\frac{1}{5}$$x2 + 6 = 2x Answer: Question 12. 3x = $$\frac{1}{4}$$x2 + 5 Answer: In Exercises 13–20, solve the equation using square roots. Question 13. s2 = 144 Answer: Question 14. a2 = 81 Answer: Question 15. (z − 6)2 = 25 Answer: Question 16. (p − 4)2 = 49 Answer: Question 17. 4(x − 1)2 + 2 = 10 Answer: Question 18. 2(x + 2)2 − 5 = 8 Answer: Question 19. $$\frac{1}{2}$$r2 − 10 = $$\frac{3}{2}$$r2 Answer: Question 20. $$\frac{1}{5}$$x2 + 2 = $$\frac{3}{5}$$x2 Answer: Question 21. ANALYZING RELATIONSHIPS Which equations have roots that are equivalent to the x-intercepts of the graph shown? A. −x2 − 6x − 8 = 0 B. 0 = (x + 2)(x + 4) C. 0 = −(x + 2)2 + 4 D. 2x2 − 4x − 6 = 0 E. 4(x + 3)2 − 4 = 0 Answer: Question 22. ANALYZING RELATIONSHIPS Which graph has x-intercepts that are equivalent to the roots of the equation (x − $$\frac{3}{2}$$)2 = $$\frac{25}{4}$$? Explain your reasoning. Answer: ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in solving the equation. Question 23. Answer: Question 24. Answer: Question 25. OPEN-ENDED Write an equation of the form x2 = d that has (a) two real solutions, (b) one real solution, and (c) no real solution. Answer: Question 26. ANALYZING EQUATIONS Which equation has one real solution? Explain. A. 3x2 + 4 = −2(x2 + 8) B. 5x2 − 4 = x2 − 4 C. 2(x + 3)2 = 18 D. $$\frac{3}{2}$$x2 − 5 = 19 Answer: In Exercises 27–34, solve the equation by factoring. Question 27. 0 = x2 + 6x + 9 Answer: Question 28. 0 = z2 − 10z + 25 Answer: Question 29. x2 − 8x = −12 Answer: Question 30. x2 − 11x = −30 Answer: Question 31. n2 − 6n = 0 Answer: Question 32. a2 − 49 = 0 Answer: Question 33. 2w2 − 16w = 12w − 48 Answer: Question 34. −y + 28 + y2 = 2y + 2y2 Answer: MATHEMATICAL CONNECTIONS In Exercises 35–38, find the value of x. Question 35. Area of rectangle = 36 Answer: Question 36. Area of circle = 25π Answer: Question 37. Area of triangle = 42 Answer: Question 38. Area of trapezoid = 32 Answer: In Exercises 39–46, solve the equation using any method. Explain your reasoning. Question 39. u2 = −9u Answer: Question 40. $$\frac{t^{2}}{20}$$ + 8 = 15 Answer: Question 41. −(x + 9)2 = 64 Answer: Question 42. −2(x + 2)2 = 5 Answer: Question 43. 7(x − 4)2 − 18 = 10 Answer: Question 44. t2 + 8t + 16 = 0 Answer: Question 45. x2 + 3x + $$\frac{5}{4}$$ = 0 Answer: Question 46. x2 − 1.75 = 0.5 Answer: In Exercises 47–54, find the zero(s) of the function. Question 47. g(x) = x2 + 6x + 8 Answer: Question 48. f(x) = x2 − 8x + 16 Answer: Question 49. h(x) = x2 + 7x − 30 Answer: Question 50. g(x) = x2 + 11x Answer: Question 51. f(x) = 2x2 − 2x − 12 Answer: Question 52. f(x) = 4x2 − 12x + 9 Answer: Question 53. g(x) = x2 + 22x + 121 Answer: Question 54. h(x) = x2 + 19x + 84 Answer: Question 55. REASONING Write a quadratic function in the form f(x) = x2 + bx + c that has zeros 8 and 11. Answer: Question 56. NUMBER SENSE Write a quadratic equation in standard form that has roots equidistant from 10 on the number line. Answer: Question 57. PROBLEM SOLVING A restaurant sells 330 sandwiches each day. For each 0.25 decrease in price, the restaurant sells about 15 more sandwiches. How much should the restaurant charge to maximize daily revenue? What is the maximum daily revenue? Answer: Question 58. PROBLEM SOLVING An athletic store sells about 200 pairs of basketball shoes per month when it charges 120 per pair. For each 2 increase in price, the store sells two fewer pairs of shoes. How much should the store charge to maximize monthly revenue? What is the maximum monthly revenue? Answer: Revenue = (120+2x)(200-2x) Revenue formula Set 120+2x = 0 and 200-2x = 0 Find the zeros of the equation 3 p = -60 and q = 100 f(x) = (x-p)(x-q) = 0 maximum at x = 20 Charge = 160 Charge = 120+2(20) = 160 Maximum Revenue = 25,600 Revenue = 160(200-2*20)= 160*160 = 25,600 Question 59. MODELING WITH MATHEMATICS Niagara Falls is made up of three waterfalls. The height of the Canadian Horseshoe Falls is about 188 feet above the lower Niagara River. A log falls from the top of Horseshoe Falls. a. Write a function that gives the height h (in feet) of the log after t seconds. How long does the log take to reach the river? b. Find and interpret h(2) − h(3). Answer: Question 60. MODELING WITH MATHEMATICS According to legend, in 1589, the Italian scientist Galileo Galilei dropped rocks of different weights from the top of the Leaning Tower of Pisa to prove his conjecture that the rocks would hit the ground at the same time. The height h (in feet) of a rock after t seconds can be modeled by h(t) = 196 − 16t2. a. Find and interpret the zeros of the function. Then use the zeros to sketch the graph. b. What do the domain and range of the function represent in this situation? Answer: Question 61. PROBLEM SOLVING You make a rectangular quilt that is 5 feet by 4 feet. You use the remaining 10 square feet of fabric to add a border of uniform width to the quilt. What is the width of the border? Answer: Question 62. MODELING WITH MATHEMATICS You drop a seashell into the ocean from a height of 40 feet. Write an equation that models the height h (in feet) of the seashell above the water after t seconds. How long is the seashell in the air? Answer: Question 63. WRITING The equation h = 0.019s2 models the height h (in feet) of the largest ocean waves when the wind speed is s knots. Compare the wind speeds required to generate 5-foot waves and 20-foot waves. Answer: Question 64. CRITICAL THINKING Write and solve an equation to find two consecutive odd integers whose product is 143. Answer: Question 65. MATHEMATICAL CONNECTIONS A quadrilateral is divided into two right triangles as shown in the figure. What is the length of each side of the quadrilateral? Answer: Question 66. ABSTRACT REASONING Suppose the equation ax2 + bx + c = 0 has no real solution and a graph of the related function has a vertex that lies in the second quadrant. a. Is the value of a positive or negative? Explain your reasoning. b. Suppose the graph is translated so the vertex is in the fourth quadrant. Does the graph have any x-intercepts? Explain. Answer: Question 67. REASONING When an object is dropped on any planet, its height h (in feet) after t seconds can be modeled by the function h = −$$\frac{g}{2}$$t2 + h0, where h0 is the object’s initial height and g is the planet’s acceleration due to gravity. Suppose a rock is dropped from the same initial height on the three planets shown. Make a conjecture about which rock will hit the ground first. Justify your answer. Answer: Question 68. PROBLEM SOLVING A café has an outdoor, rectangular patio. The owner wants to add 329 square feet to the area of the patio by expanding the existing patio as shown. Write and solve an equation to find the value of x. By what distance should the patio be extended? Answer: Question 69. PROBLEM SOLVING A flea can jump very long distances. The path of the jump of a flea can be modeled by the graph of the function y = −0.189x2 + 2.462x, where x is the horizontal distance (in inches) and y is the vertical distance (in inches). Graph the function. Identify the vertex and zeros and interpret their meanings in this situation. Answer: Question 70. HOW DO YOU SEE IT? An artist is painting a mural and drops a paintbrush. The graph represents the height h (in feet) of the paintbrush after t seconds. a. What is the initial height of the paintbrush? b. How long does it take the paintbrush to reach the ground? Explain. Answer: Question 71. MAKING AN ARGUMENT Your friend claims the equation x2 + 7x =−49 can be solved by factoring and has a solution of x = 7. You solve the equation by graphing the related function and claim there is no solution. Who is correct? Explain. Answer: Question 72. ABSTRACT REASONING Factor the expressions x2 − 4 and x2 − 9. Recall that an expression in this form is called a difference of two squares. Use your answers to factor the expression x2 − a2. Graph the related function y = x2 − a2. Label the vertex, x-intercepts, and axis of symmetry. Answer: Question 73. DRAWING CONCLUSIONS Consider the expression x2 + a2, where a > 0. a. You want to rewrite the expression as (x + m)(x + n). Write two equations that m and n must satisfy. b. Use the equations you wrote in part (a) to solve for m and n. What can you conclude? Answer: Question 74. THOUGHT PROVOKING You are redesigning a rectangular raft. The raft is 6 feet long and 4 feet wide. You want to double the area of the raft by adding to the existing design. Draw a diagram of the new raft. Write and solve an equation you can use to find the dimensions of the new raft. Answer: Question 75. MODELING WITH MATHEMATICS A high school wants to double the size of its parking lot by expanding the existing lot as shown. By what distance x should the lot be expanded? Answer: Maintaining Mathematical Proficiency Find the sum or difference. Question 76. (x2 + 2) + (2x2 − x) Answer: Question 77. (x3 + x2 − 4) + (3x2 + 10) Answer: Question 78. (−2x + 1) − (−3x2 + x) Answer: Question 79. (−3x3 + x2 − 12x) − (−6x2 + 3x − 9) Answer: Find the product. Question 80. (x + 2)(x − 2) Answer: Question 81. 2x(3 − x + 5x2) Answer: Question 82. (7 − x)(x − 1) Answer: (7x)(x1)=7x+7(−1)+(x)x+(x)(−1)=7x7x2+x=x2+(7x+x)7=x2+8x7Use (1)SimplifyGroup like terms Reduce like terms Question 83. 11x(−4x2 + 3x + 8) Answer: Lesson 3.2 Complex Numbers Essential Question What are the subsets of the set of complex numbers? In your study of mathematics, you have probably worked with only real numbers, which can be represented graphically on the real number line. In this lesson, the system of numbers is expanded to include imaginary numbers. The real numbers and imaginary numbers compose the set of complex numbers. EXPLORATION 1 Classifying Numbers Work with a partner. Determine which subsets of the set of complex numbers contain each number. a. $$\sqrt{9}$$ b. $$\sqrt{0}$$ c. −$$\sqrt{4}$$ d. $$\sqrt{\frac{4}{9}}$$ e. $$\sqrt{2}$$ f. $$\sqrt{-1}$$ EXPLORATION 2 Complex Solutions of Quadratic Equations Work with a partner. Use the definition of the imaginary unit i to match each quadratic equation with its complex solution. Justify your answers. a. x2 − 4 = 0 b. x2 + 1 = 0 c. x2 − 1 = 0 d. x2 + 4 = 0 e. x2 − 9 = 0 f. x2 + 9 = 0 A. i B. 3i C. 3 D. 2i E. 1 F. 2 Communicate Your Answer Question 3. What are the subsets of the set of complex numbers? Give an example of a number in each subset. Answer: Question 4. Is it possible for a number to be both whole and natural? natural and rational? rational and irrational? real and imaginary? Explain your reasoning. Answer: Monitoring Progress Find the square root of the number. Question 1. $$\sqrt{-4}$$ Answer: Question 2. $$\sqrt{-12}$$ Answer: Question 3. −$$\sqrt{-36}$$ Answer: Question 4. 2$$\sqrt{-54}$$ Answer: Find the values of x and y that satisfy the equation. Question 5. x + 3i = 9 − yi Answer: Question 6. 9 + 4yi = −2x + 3i Answer: Question 7. WHAT IF? In Example 4, what is the impedance of the circuit when the capacitor is replaced with one having a reactance of 7 ohms? Answer: Perform the operation. Write the answer in standard form. Question 8. (9 − i ) + (−6 + 7i ) Answer: Question 9. (3 + 7i ) − (8 − 2i ) Answer: Question 10. −4 − (1 + i) − (5 + 9i) Answer: Question 11. (−3i)(10i) Answer: Question 12. i(8 − i) Answer: Question 13. (3 + i)(5 −i) Answer: Solve the equation. Question 14. x2 = −13 Answer: Question 15. x2= −38 Answer: Question 16. x2 + 11 = 3 Answer: Question 17. x2 − 8 = −36 Answer: Question 18. 3x2 − 7 = −31 Answer: Question 19. 5x2 + 33 = 3 Answer: Find the zeros of the function. Question 20. f(x) = x2 + 7 Answer: Question 21. f(x) = −x2 − 4 Answer: Question 22. f(x) = 9x2 + 1 Answer: Complex Numbers 3.2 Exercises Vocabulary and Core Concept Check Question 1. VOCABULARY What is the imaginary unit i defined as and how can you use i? Answer: Question 2. COMPLETE THE SENTENCE For the complex number 5 + 2i, the imaginary part is ____ and the real part is ____. Answer: Question 3. WRITING Describe how to add complex numbers. Answer: Question 4. WHICH ONE DOESN’T BELONG? Which number does not belong with the other three? Explain your reasoning. Answer: Monitoring Progress and Modeling with Mathematics In Exercises 5–12, find the square root of the number. Question 5. $$\sqrt{-36}$$ Answer: Question 6. $$\sqrt{-64}$$ Answer: Question 7. $$\sqrt{-18}$$ Answer: Question 8. $$\sqrt{-24}$$ Answer: Question 9. 2$$\sqrt{-16}$$ Answer: Question 10. −3$$\sqrt{-49}$$ Answer: Question 11. −4$$\sqrt{-32}$$ Answer: Question 12. 6$$\sqrt{-63}$$ Answer: In Exercises 13–20, find the values of x and y that satisfy the equation. Question 13. 4x + 2i = 8 + yi Answer: Question 14. 3x + 6i = 27 + yi Answer: Question 15. −10x + 12i = 20 + 3yi Answer: Question 16. 9x − 18i = −36 + 6yi Answer: Question 17. 2x − yi = 14 + 12i Answer: Question 18. −12x + yi = 60 − 13i Answer: Question 19. 54 − $$\frac{1}{7}$$yi = 9x− 4i Answer: Question 20. 15 − 3yi = $$\frac{1}{2}$$x + 2i Answer: In Exercises 21–30, add or subtract. Write the answer in standard form. Question 21. (6 − i) + (7 + 3i) Answer: Question 22. (9 + 5i) + (11 + 2i ) Answer: Question 23. (12 + 4i) − (3 − 7i) Answer: Question 24. (2 − 15i) − (4 + 5i) Answer: Question 25. (12 − 3i) + (7 + 3i) Answer: Question 26. (16 − 9i) − (2 − 9i) Answer: Question 27. 7 − (3 + 4i) + 6i Answer: Question 28. 16 − (2 − 3i) − i Answer: Question 29. −10 + (6 − 5i) − 9i Answer: Question 30. −3 + (8 + 2i) + 7i Answer: Question 31. USING STRUCTURE Write each expression as a complex number in standard form. a. $$\sqrt{-9}+\sqrt{-4}-\sqrt{16}$$ b. $$\sqrt{-16}+\sqrt{8}+\sqrt{-36}$$ Answer: Question 32. REASONING The additive inverse of a complex number z is a complex number za such that z + za = 0. Find the additive inverse of each complex number. a. z = 1 + i b. z = 3 − i c. z = −2 + 8i Answer: In Exercises 33–36, find the impedance of the series circuit. Question 33. Answer: Question 35. Answer: In Exercises 37–44, multiply. Write the answer in standard form. Question 37. 3i(−5 + i) Answer: Question 38. 2i(7 − i) Answer: Question 39. (3 − 2i)(4 + i) Answer: Question 40. (7 + 5i)(8 − 6i) Answer: Question 41. (4 − 2i)(4 + 2i) Answer: Question 42. (9 + 5i)(9 − 5i) Answer: Question 43. (3 − 6i)2 Answer: Question 44. (8 + 3i)2 Answer: JUSTIFYING STEPS In Exercises 45 and 46, justify each step in performing the operation. Question 45. 11 − (4 + 3i) + 5i Answer: Question 46. (3 + 2i)(7 − 4i) Answer: REASONING In Exercises 47 and 48, place the tiles in the expression to make a true statement. Question 47. (____ − ____i) – (____ − ____i ) = 2 − 4i Answer: Question 48. ____i(____ + ____i ) = −18 − 10i Answer: In Exercises 49–54, solve the equation. Check your solution(s). Question 49. x2 + 9 = 0 Answer: Question 50. x2 + 49 = 0 Answer: Question 51. x2 − 4 = −11 Answer: Question 52. x2 − 9 = −15 Answer: Question 53. 2x2 + 6 = −34 Answer: Question 54. x2 + 7 = −47 Answer: In Exercises 55–62, find the zeros of the function. Question 55. f(x) = 3x2 + 6 Answer: Question 56. g(x) = 7x2 + 21 Answer: Question 57. h(x) = 2x2 + 72 Answer: Question 58. k(x) = −5x2 − 125 Answer: Question 59. m(x) = −x2 − 27 Answer: Question 60. p(x) = x2 + 98 Answer: Question 61. r(x) = − $$\frac{1}{2}$$x2 − 24 Answer: Question 62. f(x) = −$$\frac{1}{5}$$x2 − 10 Answer: ERROR ANALYSIS In Exercises 63 and 64, describe and correct the error in performing the operation and writing the answer in standard form. Question 63. Answer: Question 64. Answer: Question 65. NUMBER SENSE Simplify each expression. Then classify your results in the table below. a. (−4 + 7i) + (−4 − 7i) b. (2 − 6i) − (−10 + 4i) c. (25 + 15i) − (25 − 6i) d. (5 + i)(8 − i) e. (17 − 3i) + (−17 − 6i) f. (−1 + 2i)(11 − i) g. (7 + 5i) + (7 − 5i) h. (−3 + 6i) − (−3 − 8i) Answer: Question 66. MAKING AN ARGUMENT The Product Property ofSquare Roots states $$\sqrt{a}$$ • $$\sqrt{b}$$ = $$\sqrt{ab}$$ . Your friend concludes $$\sqrt{-4}$$ • $$\sqrt{-9}$$ = $$\sqrt{36}$$ = 6. Is your friend correct? Explain. Answer: Question 67. FINDING A PATTERN Make a table that shows the powers of i from i1 to i8 in the first row and the simplified forms of these powers in the second row. Describe the pattern you observe in the table. Verify the pattern continues by evaluating the next four powers of i. Answer: Question 68. HOW DO YOU SEE IT? The graphs of three functions are shown. Which function(s) has real zeros? imaginary zeros? Explain your reasoning. Answer: In Exercises 69–74, write the expression as a complex number in standard form. Question 69. (3 + 4i) − (7 − 5i) + 2i(9 + 12i) Answer: Question 70. 3i(2 + 5i) + (6 − 7i) − (9 + i) Answer: Question 71. (3 + 5i)(2 − 7i4) Answer: Question 72. 2i3(5 − 12i ) Answer: Question 73. (2 + 4i5) + (1 − 9i6) − (3 +i7) Answer: Question 74. (8 − 2i4) + (3 − 7i8) − (4 + i9) Answer: Question 75. OPEN-ENDED Find two imaginary numbers whose sum and product are real numbers. How are the imaginary numbers related? Answer: Question 76. COMPARING METHODS Describe the two different methods shown for writing the complex expression in standard form. Which method do you prefer? Explain. Answer: Question 77. CRITICAL THINKING Determine whether each statement is true or false. If it is true, give an example. If it is false, give a counterexample. a. The sum of two imaginary numbers is an imaginary number. b. The product of two pure imaginary numbers is a real number. c. A pure imaginary number is an imaginary number. d. A complex number is a real number. Answer: Question 78. THOUGHT PROVOKING Create a circuit that has an impedance of 14 − 3i. Answer: Maintaining Mathematical Proficiency Determine whether the given value of x is a solution to the equation. Question 79. 3(x − 2) + 4x − 1 = x − 1; x = 1 Answer: Question 80. x3 − 6 = 2x2 + 9 − 3x; x = −5 Answer: Question 81. −x2 + 4x = 19 — 3x2; x = −$$\frac{3}{4}$$ Answer: Write a quadratic function in vertex form whose graph is shown. Question 82. Answer: Question 83. Answer: Question 84. Answer: Lesson 3.3 Completing the Square Essential Question How can you complete the square for a quadratic expression? EXPLORATION 1 Using Algebra Tiles to Complete the Square Work with a partner. Use algebra tiles to complete the square for the expression x2 + 6x. a. You can model x2 + 6x using one x2-tile and six x-tiles. Arrange the tiles in a square. Your arrangement will be incomplete in one of the corners. b. How many 1-tiles do you need to complete the square? c. Find the value of c so that the expression x2 + 6x + c is a perfect square trinomial. d. Write the expression in part (c) as the square of a binomial. EXPLORATION 2 Drawing Conclusions Work with a partner. a. Use the method outlined in Exploration 1 to complete the table. b. Look for patterns in the last column of the table. Consider the general statement x2 + bx + c = (x + d)2. How are d and b related in each case? How are c and d related in each case? c. How can you obtain the values in the second column directly from the coefficients of x in the first column? Communicate Your Answer Question 3. How can you complete the square for a quadratic expression? Answer: Question 4. Describe how you can solve the quadratic equation x2 + 6x = 1 by completing the square. Answer: Monitoring Progress Solve the equation using square roots. Check your solution(s). Question 1. x2 + 4x + 4 = 36 Answer: Question 2. x2 − 6x + 9 = 1 Answer: Question 3. x2 − 22x + 121 = 81 Answer: Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. Question 4. x2 + 8x + c Answer: Question 5. x2 − 2x + c Answer: Question 6. x2 − 9x + c Answer: Solve the equation by completing the square. Question 7. x2 − 4x + 8 = 0 Answer: Question 8. x2 + 8x − 5 = 0 Answer: Question 9. −3x2 − 18x − 6 = 0 Answer: Question 10. 4x2 + 32x = −68 Answer: Question 11. 6x(x + 2) = −42 Answer: Question 12. 2x(x − 2) = 200 Answer: Write the quadratic function in vertex form. Then identify the vertex. Question 13. y = x2 − 8x + 18 Answer: Question 14. y = x2 + 6x + 4 Answer: Question 15. y = x2 − 2x − 6 Answer: Question 16. WHAT IF? The height of the baseball can be modeled by y = −16t2 + 80t + 2. Find the maximum height of the baseball. How long does the ball take to hit the ground? Answer: Completing the Square 3.3 Exercises Vocabulary and Core Concept Check Question 1. VOCABULARY What must you add to the expression x2 + bx to complete the square? Answer: Question 2. COMPLETE THE SENTENCE The trinomial x2 − 6x + 9 is a ____ because it equals ____. Answer: Monitoring Progress and Modeling with Mathematics In Exercises 3–10, solve the equation using square roots. Check your solution(s). Question 3. x2 − 8x + 16 = 25 Answer: Question 4. r2 − 10r + 25 = 1 Answer: Question 5. x2 − 18x + 81 = 5 Answer: Question 6. m2 + 8m + 16 = 45 Answer: Question 7. y2 − 24y + 144 = −100 Answer: Question 8. x2 − 26x + 169 = −13 Answer: Question 9. 4w2 + 4w + 1 = 75 Answer: Question 10. 4x2 − 8x + 4 = 1 Answer: In Exercises 11–20, find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. Question 11. x2 + 10x + c Answer: Question 12. x2 + 20x + c Answer: Question 13. y2 − 12y + c Answer: Question 14. t2 − 22t + c Answer: Question 15. x2 − 6x + c Answer: Question 16. x2 + 24x + c Answer: Question 17. z2 − 5z + c Answer: Question 18. x2 + 9x + c Answer: Question 19. w2 + 13w + c Answer: Question 20. s2 − 26s + c Answer: In Exercises 21–24, find the value of c. Then write an expression represented by the diagram. Question 21. Answer: Question 22. Answer: Question 23. Answer: Question 24. Answer: In Exercises 25–36, solve the equation by completing the square. Question 25. x2 + 6x + 3 = 0 Answer: Question 26. s2 + 2s − 6 = 0 Answer: Question 27. x2 + 4x − 2 = 0 Answer: Question 28. t2 − 8t − 5 = 0 Answer: Question 29. z(z + 9) = 1 Answer: Question 30. x(x + 8) = −20 Answer: Question 31. 7t2 + 28t + 56 = 0 Answer: Question 32. 6r2 + 6r + 12 = 0 Answer: Question 33. 5x(x + 6) = −50 Answer: Question 34. 4w(w − 3) = 24 Answer: Question 35. 4x2 − 30x = 12 + 10x Answer: Question 36. 3s2 + 8s = 2s − 9 Answer: Question 37. ERROR ANALYSIS Describe and correct the error in solving the equation. Answer: Question 38. ERROR ANALYSIS Describe and correct the error in finding the value of c that makes the expression a perfect square trinomial. Answer: Question 39. WRITING Can you solve an equation by completing the square when the equation has two imaginary solutions? Explain. Answer: Question 40. ABSTRACT REASONING Which of the following are solutions of the equation x2 − 2ax + a2 = b2? Justify your answers. A. ab B. −a − b C. b D. a E. a − b F. a + b Answer: USING STRUCTURE In Exercises 41–50, determine whether you would use factoring, square roots, or completing the square to solve the equation. Explain your reasoning. Then solve the equation. Question 41. x2 − 4x − 21 = 0 Answer: Question 42. x2 + 13x + 22 = 0 Answer: Question 43. (x + 4)2 = 16 Answer: Question 44. (x − 7)2 = 9 Answer: Question 45. x2 + 12x + 36 = 0 Answer: Question 46. x2 − 16x + 64 = 0 Answer: Question 47. 2x2 + 4x − 3 = 0 Answer: Question 48. 3x2 + 12x + 1 = 0 Answer: Question 49. x2 − 100 = 0 Answer: Question 50. 4x2 − 20 = 0 Answer: MATHEMATICAL CONNECTIONS In Exercises 51–54, find the value of x. Question 51. Area of rectangle = 50 Answer: Question 52. Area of parallelogram = 48 Answer: Question 53. Area of triangle = 40 Answer: Question 54. Area of trapezoid = 20 Answer: In Exercises 55–62, write the quadratic function in vertex form. Then identify the vertex. Question 55. f(x) = x2 − 8x + 19 Answer: Question 56. g(x) = x2 − 4x − 1 Answer: Question 57. g(x) = x2 + 12x + 37 Answer: Question 58. h(x) = x2 + 20x + 90 Answer: Question 59. h(x) = x2 + 2x − 48 Answer: Question 60. f(x) = x2 + 6x − 16 Answer: Question 61. f(x) = x2 − 3x + 4 Answer: Question 62. g(x) = x2 + 7x + 2 Answer: Question 63. MODELING WITH MATHEMATICS While marching, a drum major tosses a baton into the air and catches it. The height h (in feet) of the baton t seconds after it is thrown can be modeled by the function h = −16t2 + 32t + 6. a. Find the maximum height of the baton. b. The drum major catches the baton when it is 4 feet above the ground. How long is the baton in the air? Answer: Question 64. MODELING WITH MATHEMATICS A firework explodes when it reaches its maximum height. The height h (in feet) of the firework t seconds after it is launched can be modeled by h = $$-\frac{500}{9} t^{2}+\frac{1000}{3} t$$ + 10. What is the maximum height of the firework? How long is the firework in the air before it explodes? Answer: Question 65. COMPARING METHODS A skateboard shop sells about 50 skateboards per week when the advertised price is charged. For each 1 decrease in price, one additional skateboard per week is sold. The shop’s revenue can be modeled by y = (70 − x)(50 + x). a. Use the intercept form of the function to find the maximum weekly revenue. b. Write the function in vertex form to find the maximum weekly revenue. c. Which way do you prefer? Explain your reasoning. Answer: Question 66. HOW DO YOU SEE IT? The graph of the function f(x) = (x − h)2 is shown. What is the x-intercept? Explain your reasoning. Answer: Question 67. WRITING At Buckingham Fountain in Chicago, the height h (in feet) of the water above the main nozzle can be modeled by h = −162 + 89.6t, where t is the time (in seconds) since the water has left the nozzle. Describe three different ways you could find the maximum height the water reaches. Then choose a method and find the maximum height of the water. Answer: Question 68. PROBLEM SOLVING A farmer is building a rectangular pen along the side of a barn for animals. The barn will serve as one side of the pen. The farmer has 120 feet of fence to enclose an area of 1512 square feet and wants each side of the pen to be at least 20 feet long. a. Write an equation that represents the area of the pen. b. Solve the equation in part (a) to find the dimensions of the pen. Answer: Question 69. MAKING AN ARGUMENT Your friend says the equation x2 + 10x = −20 can be solved by either completing the square or factoring. Is your friend correct? Explain. Answer: Question 70. THOUGHT PROVOKING Write a function g in standard form whose graph has the same x-intercepts as the graph of f(x) = 2x2 + 8x + 2. Find the zeros of each function by completing the square. Graph each function. Answer: Question 71. CRITICAL THINKING Solve x2 + bx + c = 0 by completing the square. Your answer will be an expression for x in terms of b and c. Answer: Question 72. DRAWING CONCLUSIONS In this exercise, you will investigate the graphical effect of completing the square. a. Graph each pair of functions in the same coordinate plane. y = x2 + 2x y = x2 − 6x y = (x + 1)2 y = (x − 3)2 b. Compare the graphs of y = x2 + bx and y = (x + $$\frac{b}{2}$$)2. Describe what happens to the graph of y = x2 + bx when you complete the square. Answer: Question 73. MODELING WITH MATHEMATICS In your pottery class, you are given a lump of clay with a volume of 200 cubic centimeters and are asked to make a cylindrical pencil holder. The pencil holder should be 9 centimeters high and have an inner radius of 3 centimeters. What thickness x should your pencil holder have if you want to use all of the clay? Answer: Maintaining Mathematical Proficiency Solve the inequality. Graph the solution. Question 74. 2x − 3 < 5 Answer: Question 75. 4 − 8y ≥ 12 Answer: Question 76. $$\frac{n}{3}$$ + 6 > 1 Answer: Question 77. −$$\frac{2s}{5}$$ ≤ 8 Answer: Graph the function. Label the vertex, axis of symmetry, and x-intercepts. Question 78. g(x) = 6(x − 4)2 Answer: Question 79. h(x) = 2x(x − 3) Answer: Question 80. f(x) = x2 + 2x + 5 Answer: Question 81. f(x) = 2(x + 10)(x − 12) Answer: Quadratic Equations and Complex Numbers Study Skills: Creating a Positive Study Environment 3.1–3.3 What Did You Learn? Core Vocabulary Core Concepts Mathematical Practices Question 1. Analyze the givens, constraints, relationships, and goals in Exercise 61 on page 101. Answer: Question 2. Determine whether it would be easier to find the zeros of the function in Exercise 63 on page 117 or Exercise 67 on page 118. Answer: Study Skills: Creating a Positive Study Environment • Set aside an appropriate amount of time for reviewing your notes and the textbook, reworking your notes, and completing homework. • Set up a place for studying at home that is comfortable, but not too comfortable. The place needs to be away from all potential distractions. • Form a study group. Choose students who study well together, help out when someone misses school, and encourage positive attitudes. Quadratic Equations and Complex Numbers 3.1–3.3 Quiz Solve the equation by using the graph. Check your solution(s). Question 1. x2 − 10x + 25 = 0 Answer: Question 2. 2x2 + 16 = 12x Answer: Question 3. x2 = −2x + 8 Answer: Solve the equation using square roots or by factoring. Explain the reason for your choice. Question 4. 2x2 − 15 = 0 Answer: Question 5. 3x2 − x − 2 = 0 Answer: Question 6. (x + 3)2 = 8 ans; Question 7. Find the values of x and y that satisfy the equation 7x − 6i = 14 + yi. Answer: Perform the operation. Write your answer in standard form Question 8. (2 + 5i) + (−4 + 3i) Answer: Question 9. (3 + 9i) − (1 − 7i) Answer: Question 10. (2 + 4i)(−3 − 5i) Answer: Question 11. Find the zeros of the function f(x) = 9x2 + 2. Does the graph of the function intersect the x-axis? Explain your reasoning. Answer: Solve the equation by completing the square. Question 12. x2 − 6x + 10 = 0 Answer: Question 13. x2 + 12x + 4 = 0 Answer: Question 14. 4x(x + 6) = −40 Answer: Question 15. Write y = x2 − 10x + 4 in vertex form. Then identify the vertex. Answer: Question 16. A museum has a café with a rectangular patio. The museum wants to add 464 square feet to the area of the patio by expanding the existing patio as shown. a. Find the area of the existing patio. b. Write an equation to model the area of the new patio. c. By what distance x should the length of the patio be expanded? Answer: Question 17. Find the impedance of the series circuit. Answer: Question 18. The height h (in feet) of a badminton birdie t seconds after it is hit can be modeled by the function h = −16t2 + 32t + 4. a. Find the maximum height of the birdie. b. How long is the birdie in the air? Answer: Lesson 3.4 Using the Quadratic Formula Essential Question How can you derive a general formula for solving a quadratic equation? EXPLORATION 1 Deriving the Quadratic Formula Work with a partner. Analyze and describe what is done in each step in the development of the Quadratic Formula. EXPLORATION 2 Using the Quadratic Formula Work with a partner. Use the Quadratic Formula to solve each equation. a. x2 − 4x + 3 = 0 b. x2 − 2x + 2 = 0 c. x2 + 2x − 3 = 0 d. x2 + 4x + 4 = 0 e. x2 − 6x + 10 = 0 f. x2 + 4x + 6 = 0 Communicate Your Answer Question 3. How can you derive a general formula for solving a quadratic equation? Answer: Question 4. Summarize the following methods you have learned for solving quadratic equations: graphing, using square roots, factoring, completing the square, and using the Quadratic Formula. Answer: Monitoring Progress Solve the equation using the Quadratic Formula. Question 1. x2 − 6x + 4 = 0 Answer: Question 2. 2x2 + 4 = −7x Answer: Question 3. 5x2 = x + 8 Answer: Solve the equation using the Quadratic Formula. Question 4. x2 + 41 = −8x Answer: Question 5. −9x2 = 30x + 25 Answer: Question 6. 5x − 7x2 = 3x + 4 Answer: Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. Question 7. 4x2 + 8x + 4 = 0 Answer: qm 8. $$\frac{1}{2}$$x2 + x − 1 = 0 Answer: Question 9. 5x2 = 8x − 13 Answer: Question 10. 7x2 − 3x = 6 Answer: Question 11. 4x2 + 6x = −9 Answer: Question 12. −5x2 + 1 = 6 − 10x Answer: Question 13. Find a possible pair of integer values for a and c so that the equation ax2 + 3x + c = 0 has two real solutions. Then write the equation. Answer: Question 14. WHAT IF? The ball leaves the juggler’s hand with an initial vertical velocity of 40 feet per second. How long is the ball in the air? Answer: Using the Quadratic Formula 3.4 Exercises Vocabulary and Core Concept Check Question 1. COMPLETE THE SENTENCE When a, b, and c are real numbers such that a ≠ 0, the solutions of the quadratic equation ax2 + bx + c = 0 are x= ____________. Answer: Question 2. COMPLETE THE SENTENCE You can use the ____________ of a quadratic equation to determine the number and type of solutions of the equation. Answer: Question 3. WRITING Describe the number and type of solutions when the value of the discriminant is negative. Answer: Question 4. WRITING Which two methods can you use to solve any quadratic equation? Explain when you might prefer to use one method over the other. Answer: Monitoring Progress and Modeling with Mathematics In Exercises 5–18, solve the equation using the Quadratic Formula. Use a graphing calculator to check your solution(s). Question 5. x2 − 4x + 3 = 0 Answer: Question 6. 3x2 + 6x + 3 = 0 Answer: Question 7. x2 + 6x + 15 = 0 Answer: Question 8. 6x2 − 2x + 1 = 0 Answer: Question 9. x2 − 14x = −49 Answer: Question 10. 2x2 + 4x = 30 Answer: Question 11. 3x2 + 5 = −2x Answer: Question 12. −3x = 2x2 − 4 Answer: Question 13. −10x = −25 − x2 Answer: Question 14. −5x2 − 6 = −4x Answer: Question 15. −4x2 + 3x = −5 Answer: Question 16. x2 + 121 = −22x Answer: Question 17. −z2 = −12z + 6 Answer: Question 18. −7w + 6 = −4w2 Answer: In Exercises 19–26, find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. Question 19. x2 + 12x + 36 = 0 Answer: Question 20. x2 − x + 6 = 0 Answer: Question 21. 4n2 − 4n − 24 = 0 Answer: Question 22. −x2 + 2x + 12 = 0 Answer: Question 23. 4x2 = 5x − 10 Answer: Question 24. −18p = p2 + 81 Answer: Question 25. 24x = −48 − 3x2 Answer: Question 26. −2x2 − 6 = x2 Answer: Question 27. USING EQUATIONS What are the complex solutions of the equation 2x2− 16x+ 50 = 0? A. 4 + 3i, 4 − 3i B. 4 + 12i, 4 − 12i C. 16 + 3i, 16 − 3i D. 16 + 12i, 16 − 12i Answer: Question 28. USING EQUATIONS Determine the number and type of solutions to the equation x2 + 7x = −11. A. two real solutions B. one real solution C. two imaginary solutions D. one imaginary solution Answer: ANALYZING EQUATIONS In Exercises 29–32, use the discriminant to match each quadratic equation with the correct graph of the related function. Explain your reasoning. Question 29. x2 − 6x + 25 = 0 Answer: Question 30. 2x2 − 20x + 50 = 0 Answer: Question 31. 3x2 + 6x − 9 = 0 Answer: Question 32. 5x2 − 10x − 35 = 0 Answer: ERROR ANALYSIS In Exercises 33 and 34, describe and correct the error in solving the equation. Question 33. Answer: Question 34. Answer: OPEN-ENDED In Exercises 35–40, find a possible pair of integer values for a and c so that the quadratic equation has the given solution(s). Then write the equation. Question 35. ax2 + 4x + c = 0; two imaginary solutions Answer: Question 36. ax2 + 6x + c = 0; two real solutions Answer: Question 37. ax2 − 8x + c = 0; two real solutions Answer: Question 38. ax2 − 6x + c = 0; one real solution Answer: Question 39. ax2 + 10x = c; one real solution Answer: Question 40. −4x + c = −ax2; two imaginary solutions Answer: USING STRUCTURE In Exercises 41–46, use the Quadratic Formula to write a quadratic equation that has the given solutions. Question 41. x = $$\frac{-8 \pm \sqrt{-176}}{-10}$$ Answer: Question 42. x = $$\frac{15 \pm \sqrt{-215}}{22}$$ Answer: Question 43. x = $$\frac{-4 \pm \sqrt{-124}}{-14}$$ Answer: Question 44. x = $$\frac{-9 \pm \sqrt{137}}{4}$$ Answer: Question 45. x = $$\frac{-4 \pm 2}{6}$$ Answer: Question 46. x = $$\frac{2 \pm 4}{-2}$$ Answer: COMPARING METHODS In Exercises 47–58, solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do you prefer? Explain. Question 47. 3x2 − 21 = 3 Answer: Question 48. 5x2 + 38 = 3 Answer: Question 49. 2x2 − 54 = 12x Answer: Question 50. x2 = 3x + 15 Answer: Question 51. x2 − 7x + 12 = 0 Answer: Question 52. x2 + 8x − 13 = 0 Answer: Question 53. 5x2 − 50x = −135 Answer: Question 54. 8x2 + 4x + 5 = 0 Answer: Question 55. −3 = 4x2 + 9x Answer: Question 56. −31x + 56 = −x2 Answer: Question 57. x2 = 1 − x Answer: Question 58. 9x2 + 36x + 72 = 0 Answer: MATHEMATICAL CONNECTIONS In Exercises 59 and 60, find the value for x. Question 59. Area of the rectangle = 24 m2 Answer: Question 6. Area of the triangle = 8ft2 Answer: Question 61. MODELING WITH MATHEMATICS A lacrosse player throws a ball in the air from an initial height of 7 feet. The ball has an initial vertical velocity of 90 feet per second. Another player catches the ball when it is 3 feet above the ground. How long is the ball in the air? Answer: Question 62. NUMBER SENSE Suppose the quadratic equation ax2 + 5x + c = 0 has one real solution. Is it possible for a and c to be integers? rational numbers? Explain your reasoning. Then describe the possible values of a and c. Answer: Question 63. MODELING WITH MATHEMATICS In a volleyball game, a player on one team spikes the ball over the net when the ball is 10 feet above the court. The spike drives the ball downward with an initial vertical velocity of 55 feet per second. How much time does the opposing team have to return the ball before it touches the court? Answer: Question 64. MODELING WITH MATHEMATICS An archer is shooting at targets. The height of the arrow is 5 feet above the ground. Due to safety rules, the archer must aim the arrow parallel to the ground. a. How long does it take for the arrow to hit a target that is 3 feet above the ground? b. What method did you use to solve the quadratic equation? Explain. Answer: Question 65. PROBLEM SOLVING A rocketry club is launching model rockets. The launching pad is 30 feet above the ground. Your model rocket has an initial vertical velocity of 105 feet per second. Your friend’s model rocket has an initial vertical velocity of 100 feet per second. a. Use a graphing calculator to graph the equations of both model rockets. Compare the paths. b. After how many seconds is your rocket 119 feet above the ground? Explain the reasonableness of your answer(s). Answer: Question 66. PROBLEM SOLVING The number A of tablet computers sold (in millions) can be modeled by the function A = 4.5t2 + 43.5t + 17, where t represents the year after 2010. a. In what year did the tablet computer sales reach 65 million? b. Find the average rate of change from 2010 to 2012 and interpret the meaning in the context of the situation. c. Do you think this model will be accurate after a new, innovative computer is developed? Explain. Answer: Question 67. MODELING WITH MATHEMATICS A gannet is a bird that feeds on fish by diving into the water. A gannet spots a fish on the surface of the water and dives 100 feet to catch it. The bird plunges toward the water with an initial vertical velocity of −88 feet per second. a. How much time does the fish have to swim away? b. Another gannet spots the same fish, and it is only 84 feet above the water and has an initial vertical velocity of −70 feet per second. Which bird will reach the fish first? Justify your answer. Answer: Question 68. USING TOOLS You are asked to find a possible pair of integer values for a and c so that the equation ax2 − 3x + c = 0 has two real solutions. When you solve the inequality for the discriminant, you obtain ac < 2.25. So, you choose the values a = 2 and c = 1. Your graphing calculator displays the graph of your equation in a standard viewing window. Is your solution correct? Explain. Answer: Question 69. PROBLEM SOLVING Your family has a rectangular pool that measures 18 feet by 9 feet. Your family wants to put a deck around the pool but is not sure how wide to make the deck. Determine how wide the deck should be when the total area of the pool and deck is 400 square feet. What is the width of the deck? Answer: Question 70. HOW DO YOU SEE IT? The graph of a quadratic function y = ax2 + bx + c is shown. Determine whether each discriminant of ax2 + bx + c = 0 is positive, negative, or zero. Then state the number and type of solutions for each graph. Explain your reasoning. Answer: Question 71. CRITICAL THINKING Solve each absolute value equation. a. |x2 – 3x – 14| = 4 b. x2 = |x| + 6 Answer: Question 72. MAKING AN ARGUMENT The class is asked to solve the equation 4x2 + 14x + 11 = 0. You decide to solve the equation by completing the square. Your friend decides to use the Quadratic Formula. Whose method is more efficient? Explain your reasoning. Answer: Question 73. ABSTRACT REASONING For a quadratic equation ax2 + bx + c = 0 with two real solutions, show that the mean of the solutions is $$\frac{b}{2a}$$. How is this fact related to the symmetry of the graph of y = ax2 + bx + c? Answer: Question 74. THOUGHT PROVOKING Describe a real-life story that could be modeled by h = −16t2 + v0t + h0 . Write the height model for your story and determine how long your object is in the air. Answer: Question 75. REASONING Show there is no quadratic equation ax2+bx+c= 0 such that a, b, and c are real numbers and 3i and −2i are solutions. Answer: Question 76. MODELING WITH MATHEMATICS The Stratosphere Tower in Las Vegas is 921 feet tall and has a “needle” at its top that extends even higher into the air. A thrill ride called Big Shot catapults riders 160 feet up the needle and then lets them fall back to the launching pad. a. The height h (in feet) of a rider on the Big Shot can be modeled by h = −16t2 + v0 t + 921, where t is the elapsed time (in seconds) after launch and v0 is the initial vertical velocity (in feet per second). Find v0 using the fact that the maximum value of h is 921 + 160 = 1081 feet. b. A brochure for the Big Shot states that the ride up the needle takes 2 seconds. Compare this time to the time given by the model h = −16t2 + v0 t + 921, where v0 is the value you found in part (a). Discuss the accuracy of the model. Answer: Maintaining Mathematical Proficiency Solve the system of linear equations by graphing. Question 77. −x + 2y = 6 x + 4y = 24 Answer: Question 78. y = 2x − 1 y = x + 1 Answer: Question 79. 3x + y = 4 6x + 2y = −4 Answer: Question 80. y = −x + 2 −5x + 5y = 10 Answer: Graph the quadratic equation. Label the vertex and axis of symmetry. Question 81. y = −x2 + 2x + 1 Answer: Question 82. y = 2x2 − x + 3 Answer: Question 83. y = 0.5x2 + 2x + 5 Answer: Question 84. y = −3x2 − 2 Answer: Lesson 3.5 Solving Nonlinear Systems Essential Question How can you solve a nonlinear system of equations? EXPLORATION 1 Solving Nonlinear Systems of Equations Work with a partner. Match each system with its graph. Explain your reasoning. Then solve each system using the graph. EXPLORATION 2 Solving Nonlinear Systems of Equations Work with a partner. Look back at the nonlinear system in Exploration 1(f). Suppose you want a more accurate way to solve the system than using a graphical approach. a. Show how you could use a numerical approach by creating a table. For instance, you might use a spreadsheet to solve the system. b. Show how you could use an analytical approach. For instance, you might try solving the system by substitution or elimination. Communicate Your Answer Question 3. How can you solve a nonlinear system of equations? Answer: Question 4. Would you prefer to use a graphical, numerical, or analytical approach to solve the given nonlinear system of equations? Explain your reasoning. Answer: Solve the system using any method. Explain your choice of method. Question 1. y = −x2 + 4 y = −4x + 8 Answer: Question 2. x2 + 3x + y = 0 2x + y = 5 Answer: Question 3. 2x2 + 4x − y =−2 x2 + y = 2 Answer: Solve the system. Question 4. x2 + y2 = 16 y = −x + 4 Answer: Question 5. x2 + y2 = 4 y = x + 4 Answer: Question 6. x2 + y2 = 1 y = $$\frac{1}{2}$$x + $$\frac{1}{2}$$ Answer: Solve the equation by graphing. Question 7. x2 − 6x + 15 = −(x − 3)2 + 6 Answer: Question 8. (x + 4)(x − 1) = −x2 + 3x + 4 Answer: Solving Nonlinear Systems 3.5 Exercises Vocabulary and Core Concept Check Question 1. WRITING Describe the possible solutions of a system consisting of two quadratic equations. Answer: Question 2. WHICH ONE DOESN’T BELONG? Which system does not belong with the other three? Explain your reasoning. Answer: Monitoring Progress and Modeling with Mathematics In Exercises 3–10, solve the system by graphing. Check your solution(s). Question 3. y = x + 2 y = 0.5(x + 2)2 Answer: Question 4. y = (x − 3)2 + 5 y = 5 Answer: Question 5. y = $$\frac{1}{3}$$x + 2 y = −3x2 − 5x − 4 Answer: Question 6. y = −3x2 − 30x − 71 y = −3x − 17 Answer: Question 7. y = x2 + 8x + 18 y = −2x2 − 16x − 30 Answer: Question 8. y = −2x2 − 9 y = −4x − 1 Answer: Question 9. y = (x − 2)2 y = −x2 + 4x − 2 Answer: Question 10. y = $$\frac{1}{2}$$(x + 2)2 y = −$$\frac{1}{2}$$x2 + 2 Answer: In Exercises 11–14, solve the system of nonlinear equations using the graph. Question 11. Answer: Question 12. Answer: We watch the graphs of the two functions carefully. The two graphs represent two parabolas: one opens up and the other opens down. They do not intersect, so the system has no solution. Question 13. Answer: Question 14. Answer: In Exercises 15–24, solve the system by substitution. Question 15. y = x + 5 y = x2 − x + 2 Answer: Question 16. x2 + y2 = 49 y = 7 − x Answer: Question 17. x2 + y2 = 64 y = −8 Answer: Question 18. x = 3 −3x2 + 4x − y = 8 Answer: Question 19. 2x2 + 4x − y = −3 −2x + y = −4 Answer: Question 20. 2x − 3 = y + 5x2 y = −3x − 3 Answer: Question 21. y = x2 − 1 −7 = −x2 − y Answer: Question 22. y + 16x − 22 = 4x2 4x2 − 24x + 26 + y = 0 Answer: Question 23. x2 + y2 = 7 x + 3y = 21 Answer: Question 24. x2 + y2 = 5 −x + y = −1 Answer: Question 25. USING EQUATIONS Which ordered pairs are solutions of the nonlinear system? y = $$\frac{1}{2}$$x2 − 5x + $$\frac{21}{2}$$ y = −$$\frac{1}{2}$$x + $$\frac{13}{2}$$ A. (1, 6) B. (3, 0) C. (8, 2.5) D. (7, 0) Answer: Question 26. USING EQUATIONS How many solutions does the system have? Explain your reasoning. y = 7x2 − 11x + 9 y = −7x2 + 5x − 3 A. 0 B. 1 C. 2 D. 4 Answer: In Exercises 27–34, solve the system by elimination. Question 27. 2x2 − 3x −y =−5 −x + y = 5 Answer: Question 28. −3x2 + 2x − 5 = y −x + 2 = −y Answer: Question 29. −3x2 + y = −18x + 29 −3x2 − y = 18x − 25 Answer: Question 30. y = −x2 − 6x 10 y = 3x2 + 18x + 22 Answer: Question 31. y + 2x = −14 −x2 − y − 6x = 11 Answer: Question 32. y = x2 + 4x + 7 −y = 4x + 7 Answer: Question 33. y = −3x2 − 30x − 76 y = 2x2 + 20x + 44 Answer: Question 34. −10x2 + y = −80x + 155 5x2 + y = 40x − 85 Answer: Question 35. ERROR ANALYSIS Describe and correct the error in using elimination to solve a system. Answer: Question 36. NUMBER SENSE The table shows the inputs and outputs of two quadratic equations. Identify the solution(s) of the system. Explain your reasoning. Answer: In Exercises 37–42, solve the system using any method. Explain your choice of method. Question 37. y = x2 − 1 −y = 2x2 + 1 Answer: Question 38. y = −4x2 − 16x − 13 −3x2 + y + 12x = 17 Answer: Question 39. −2x + 10 + y = $$\frac{1}{3}$$x2 y = 10 Answer: Question 40. y = 0.5x2 − 10 y = −x2 + 14 Answer: Question 41. y = −3(x − 4)2 + 6 (x − 4)2 + 2 − y = 0 Answer: Question 42. −x2 + y2 = 100 y = −x + 14 Answer: USING TOOLS In Exercises 43–48, solve the equation by graphing. Question 43. x2 + 2x = −$$\frac{1}{2}$$x2 + 2x Answer: Question 44. 2x2 − 12x − 16 = −6x2 + 60x − 144 Answer: Question 45. (x + 2)(x − 2) = −x2 + 6x − 7 Answer: Question 46. −2x2 − 16x − 25 = 6x2 + 48x + 95 Answer: Question 47. (x − 2)2 − 3 = (x + 3)(−x + 9) − 38 Answer: Question 48. (−x + 4)(x + 8) − 42 = (x + 3)(x + 1) − 1 Answer: Question 49. REASONING A nonlinear system contains the equations of a constant function and a quadratic function. The system has one solution. Describe the relationship between the graphs. Answer: Question 50. PROBLEM SOLVING The range (in miles) of a broadcast signal from a radio tower is bounded by a circle given by the equation x2 + y2= 1620. A straight highway can be modeled by the equation y = −$$\frac{1}{3}$$x + 30. For what lengths of the highway are cars able to receive the broadcast signal? Answer: Question 51. PROBLEM SOLVING A car passes a parked police car and continues at a constant speed r. The police car begins accelerating at a constant rate when it is passed. The diagram indicates the distance d (in miles) the police car travels as a function of time t (in minutes) after being passed. Write and solve a system of equations to find how long it takes the police car to catch up to the other car. Answer: Question 52. THOUGHT PROVOKING Write a nonlinear system that has two different solutions with the same y-coordinate. Sketch a graph of your system. Then solve the system. Answer: Question 53. OPEN-ENDED Find three values for m so the system has no solution, one solution, and two solutions. Justify your answer using a graph. 3y = −x2 + 8x − 7 y = mx + 3 Answer: Question 54. MAKING AN ARGUMENT You and a friend solve the system shown and determine that x = 3 and x = −3. You use Equation 1 to obtain the solutions (3, -3), (3, −3), (−3, 3), and (−3, −3). Your friend uses Equation 2 to obtain the solutions (3, 3) and (−3, −3). Who is correct? Explain your reasoning. x2 + y2 = 18 Equation 1 x − y = 0 Equation 2 Answer: Question 55. COMPARING METHODS Describe two different ways you could solve the quadratic equation. Which way do you prefer? Explain your reasoning. −2x2 + 12x − 17 = 2x2 − 16x + 31 Answer: Question 56. ANALYZING RELATIONSHIPS Suppose the graph of a line that passes through the origin intersects the graph of a circle with its center at the origin. When you know one of the points of intersection, explain how you can find the other point of intersection without performing any calculations. Answer: Question 57. WRITING Describe the possible solutions of a system that contains (a) one quadratic equation and one equation of a circle, and (b) two equations of circles. Sketch graphs to justify your answers. Answer: Question 58. HOW DO YOU SEE IT? The graph of a nonlinear system is shown. Estimate the solution(s). Then describe the transformation of the graph of the linear function that results in a system with no solution. Answer: Question 59. MODELING WITH MATHEMATICS To be eligible for a parking pass on a college campus, a student must live at least 1 mile from the campus center. a. Write equations that represent the circle and Oak Lane. b. Solve the system that consists of the equations in part (a). c. For what length of Oak Lane are students not eligible for a parking pass? Answer: Question 60. CRITICAL THINKING Solve the system of three equations shown. x2 + y2 = 4 2y = x2 − 2x + 4 y = −x + 2 Answer: Maintaining Mathematical Proficiency Solve the inequality. Graph the solution on a number line. Question 61. 4x − 4 > 8 Answer: Question 62. −x + 7 ≤ 4 − 2x Answer: Question 63. −3(x − 4) ≥ 24 Answer: Write an inequality that represents the graph. Question 64. Answer: Question 65. Answer: Question 66. Answer: Lesson 3.6 Quadratic Inequalities Essential Question How can you solve a quadratic inequality? EXPLORATION 1 Solving a Quadratic Inequality Work with a partner. The graphing calculator screen shows the graph of f(x) = x2 + 2x − 3. Explain how you can use the graph to solve the inequality x2 + 2x − 3 ≤ 0. Then solve the inequality. EXPLORATION 2 Solving Quadratic Inequalities Work with a partner. Match each inequality with the graph of its related quadratic function. Then use the graph to solve the inequality. a. x2 − 3x + 2 > 0 b. x2 − 4x + 3 ≤ 0 c. x2 − 2x − 3 < 0 d. x2 + x − 2 ≥ 0 e. x2 − x − 2 < 0 f. x2 − 4 > 0 Communicate Your Answer Question 3. How can you solve a quadratic inequality? Answer: Question 4. Explain how you can use the graph in Exploration 1 to solve each inequality. Then solve each inequality. Answer: Monitoring Progress Graph the inequality. Question 1. y ≥ x2 + 2x − 8 Answer: Question 2. y ≤ 2x2 −x − 1 Answer: Question 3. y > −x2 + 2x + 4 Answer: Question 4. Graph the system of inequalities consisting of y ≤ −x2 and y > x2 − 3. Answer: Solve the inequality. Question 5. 2x2 + 3x ≤ 2 Answer: Question 6. −3x2 − 4x + 1 < 0 Answer: Question 7. 2x2 + 2 > −5x Answer: Question 8. WHAT IF? In Example 6, the area must be at least 8500 square feet. Describe the possible lengths of the parking lot. Answer: Quadratic Inequalities 3.6 Exercises Vocabulary and Core Concept Check Question 1. WRITING Compare the graph of a quadratic inequality in one variable to the graph of a quadratic inequality in two variables. Answer: Question 2. WRITING Explain how to solve x2 + 6x − 8 < 0 using algebraic methods and using graphs. Answer: Monitoring Progress and Modeling with Mathematics In Exercises 3–6, match the inequality with its graph. Explain your reasoning. Question 3. y ≤ x2 + 4x + 3 Answer: Question 4. y > −x2 + 4x − 3 Answer: Question 5. y < x2 − 4x + 3 Answer: Question 6. y ≥ x2 + 4x + 3 Answer: In Exercises 7–14, graph the inequality. Question 7. y < −x2 Answer: Question 8. y ≥ 4x2 Answer: Question 9. y > x2 − 9 Answer: Question 10. y < x2 + 5 Answer: Question 11. y ≤ x2 + 5x Answer: Question 12. y ≥ −2x2 + 9x − 4 Answer: graph y≥-2x²+9x-4 X= – b/2a X= – 9/-4 = 2.25 y = -2(2.25)² + 9(2.25) – 4 y = -2(5.0625)+20.25 – 4 y = -10.125 +20.25 – 4 y = 6.125 or 49/8 V(2.25,6.125) Question 13. y > 2(x + 3)2 − 1 Answer: Question 14. y ≤ (x − $$\frac{1}{2}$$)2 + $$\frac{5}{2}$$ Answer: ANALYZING RELATIONSHIPS In Exercises 15 and 16, use the graph to write an inequality in terms of f (x) so point P is a solution. Question 15. Answer: Question 16. Answer: ERROR ANALYSIS In Exercises 17 and 18, describe and correct the error in graphing y ≥ x2 + 2. Question 17. Answer: Question 18. Answer: Question 19. MODELING WITH MATHEMATICS A hardwood shelf in a wooden bookcase can safely support a weight W (in pounds) provided W ≤ 115x2, where x is the thickness (in inches) of the shelf. Graph the inequality and interpret the solution. Answer: Question 20. MODELING WITH MATHEMATICS A wire rope can safely support a weight W (in pounds) provided W ≤ 8000d2, where d is the diameter (in inches) of the rope. Graph the inequality and interpret the solution. Answer: In Exercises 21–26, graph the system of quadratic inequalities. Question 21. y ≥ 2x2 y < −x2 + 1 Answer: Question 22. y > −5x2 y > 3x2 − 2 Answer: Question 23. y ≤ −x2 + 4x − 4 y < x2 + 2x − 8 Answer: Question 24. y ≥ x2 − 4 y ≤ −2x2 + 7x + 4 Answer: Question 25. y ≥ 2x2 + x − 5 y < −x2 + 5x + 10 Answer: Question 26. y ≥ x2 − 3x − 6 y ≥ x2 + 7x + 6 Answer: In Exercises 27–34, solve the inequality algebraically. Question 27. 4x2 < 25 Answer: Question 28. x2 + 10x + 9 < 0 Answer: Question 29. x2 − 11x ≥ −28 Answer: Question 30. 3x2 − 13x > −10 Answer: Question 31. 2x2 − 5x − 3 ≤ 0 Answer: Question 32. 4x2 + 8x − 21 ≥ 0 Answer: Question 33. $$\frac{1}{2}$$x2 − x > 4 Answer: Question 34. −$$\frac{1}{2}$$x2 + 4x ≤ 1 Answer: In Exercises 35–42, solve the inequality by graphing. Question 35. x2 − 3x + 1 < 0 Answer: Question 36. x2 − 4x + 2 > 0 Answer: Question 37. x2 + 8x > −7 Answer: Question 38. x2 + 6x < −3 Answer: Question 39. 3x2 − 8 ≤ − 2x Answer: Question 40. 3x2 + 5x − 3 < 1 Answer: Question 41. $$\frac{1}{3}$$x2 + 2x ≥ 2 Answer: Question 42. $$\frac{3}{4}$$x2 + 4x ≥ 3 Answer: Question 43. DRAWING CONCLUSIONS Consider the graph of the function f(x) = ax2 + bx + c. a. What are the solutions of ax2 + bx + c < 0? b. What are the solutions of ax2 + bx + c > 0? c. The graph of g represents a reflection in the x-axis of the graph of f. For which values of x is g(x) positive? Answer: Question 44. MODELING WITH MATHEMATICS A rectangular fountain display has a perimeter of 400 feet and an area of at least 9100 feet. Describe the possible widths of the fountain. Answer: Question 45. MODELING WITH MATHEMATICS The arch of the Sydney Harbor Bridge in Sydney, Australia, can be modeled by y = −0.00211x2 + 1.06x, where x is the distance (in meters) from the left pylons and y is the height (in meters) of the arch above the water. For what distances x is the arch above the road? Answer: Question 46. PROBLEM SOLVING The number T of teams that have participated in a robot-building competition for high-school students over a recent period of time x(in years) can be modeled by T(x) = 17.155x2 + 193.68x + 235.81, 0 ≤ x ≤ 6. After how many years is the number of teams greater than 1000? Justify your answer. Answer: Question 47. PROBLEM SOLVING A study found that a driver’s reaction time A(x) to audio stimuli and his or her reaction time V(x) to visual stimuli (both in milliseconds) can be modeled by A(x) = 0.0051x2 − 0.319x + 15, 16 ≤ x ≤ 70 V(x) = 0.005x2 − 0.23x + 22, 16 ≤ x ≤ 70 where x is the age (in years) of the driver. a. Write an inequality that you can use to find the x-values for which A(x) is less than V(x). b. Use a graphing calculator to solve the inequality A(x) < V(x). Describe how you used the domain 16 ≤ x ≤ 70 to determine a reasonable solution. c. Based on your results from parts (a) and (b), do you think a driver would react more quickly to a traffic light changing from green to yellow or to the siren of an approaching ambulance? Explain. Answer: Question 48. HOW DO YOU SEE IT? The graph shows a system of quadratic inequalities. a. Identify two solutions of the system. b. Are the points (1, −2) and (5, 6) solutions of the system? Explain. c. Is it possible to change the inequality symbol(s) so that one, but not both of the points in part (b), is a solution of the system? Explain. Answer: Question 49. MODELING WITH MATHEMATICS The length L (in millimeters) of the larvae of the black porgy fish can be modeled by L(x) = 0.00170x2 + 0.145x + 2.35, 0 ≤ x ≤ 40 where x is the age (in days) of the larvae. Write and solve an inequality to find at what ages a larva’s length tends to be greater than 10 millimeters. Explain how the given domain affects the solution. Answer: Question 50. MAKING AN ARGUMENT You claim the system of inequalities below, where a and b are real numbers, has no solution. Your friend claims the system will always have at least one solution. Who is correct? Explain. y < (x + a)2 y < (x + b)2 Answer: Question 51. MATHEMATICAL CONNECTIONS The area A of the region bounded by a parabola and a horizontal line can be modeled by A= $$\frac{2}{3}$$bh, where b and h are as defined in the diagram. Find the area of the region determined by each pair of inequalities. Answer: Question 52. THOUGHT PROVOKING Draw a company logo that is created by the intersection of two quadratic inequalities. Justify your answer. Answer: Question 53. REASONING A truck that is 11 feet tall and 7 feet wide is traveling under an arch. The arch can be modeled by y = −0.0625x2 + 1.25x + 5.75, where x and y are measured in feet. a. Will the truck fit under the arch? Explain. b. What is the maximum width that a truck 11 feet tall can have and still make it under the arch? c. What is the maximum height that a truck 7 feet wide can have and still make it under the arch? Answer: Maintaining Mathematical Proficiency Graph the function. Label the x-intercept(s) and the y-intercept. Question 54. f(x) = (x + 7)(x − 9) Answer: Question 55. g(x) = (x − 2)2 − 4 Answer: Question 56. h(x) = −x2 + 5x − 6 Answer: Identify the $x$ -intercepts. Rewrite the given function in Intercepts Form: \begin{align*} h(x)&=a(x-p)(x-q)\\ h(x)&=-x^2+5x-6\\ &=-x^2+2x+3x-6\\ &=-x(x-2)+3(x-2)\\ &=-(x-2)(x-3)\\ \end{align*} The $x$ -intercepts are: \begin{align*} p&=2\\ q&=3 \end{align*} Therefore the parabola passes through the points $(2,0)$ and $(3,0)$ . Find the minimum value or maximum value of the function. Then describe where the function is increasing and decreasing. Question 57. f(x) = −x2 − 6x − 10 Answer: Question 58. h(x) = $$\frac{1}{2}$$(x + 2)2 − 1 Answer: Question 59. f(x) = −(x − 3)(x + 7) Answer: Question 60. h(x) = x2 + 3x − 18 Answer: Quadratic Equations and Complex Numbers Performance Task: Algebra in Genetics: The Hardy-Weinberg Law 3.4–3.6 What Did You Learn? Core Vocabulary Core Concepts Section 3.4 Solving Equations Using the Quadratic Formula, p. 122 Analyzing the Discriminant of ax2+bx+c= 0, p. 124 Methods for Solving Quadratic Equations, p. 125 Modeling Launched Objects, p. 126 Section 3.5 Solving Systems of Nonlinear Equations, p. 132 Solving Equations by Graphing, p. 135 Section 3.6 Graphing a Quadratic Inequality in Two Variables, p. 140 Solving Quadratic Inequalities in One Variable, p. 142 Mathematical Practices Question 1. How can you use technology to determine whose rocket lands first in part (b) of Exercise 65 on page 129? Answer: Question 2. What question can you ask to help the person avoid making the error in Exercise 54 on page 138? Answer: Question 3. Explain your plan to find the possible widths of the fountain in Exercise 44 on page 145. Answer: Performance Task: Algebra in Genetics: The Hardy-Weinberg Law Some people have attached earlobes, the recessive trait. Some people have free earlobes, the dominant trait. What percent of people carry both traits? To explore the answers to this question and more, go to BigIdeasMath.com. Quadratic Equations and Complex Numbers Chapter Review 3.1 Solving Quadratic Equations (pp. 93–102) Question 1. Solve x2 − 2x − 8 = 0 by graphing. Answer: Solve the equation using square roots or by factoring. Question 2. 3x2 − 4 = 8 Answer: Question 3. x2 + 6x − 16 = 0 Answer: Question 4. 2x2 − 17x = −30 Answer: Question 5. A rectangular enclosure at the zoo is 35 feet long by 18 feet wide. The zoo wants to double the area of the enclosure by adding the same distance x to the length and width. Write and solve an equation to find the value of x. What are the dimensions of the enclosure? Answer: 3.2 Complex Numbers (pp. 103–110) Question 6. Find the values x and y that satisfy the equation 36 − yi = 4x + 3i. Answer: Perform the operation. Write the answer in standard form. Question 7. (−2 + 3i ) + (7 − 6i ) Answer: Question 8. (9 + 3i ) − (−2 − 7i ) Answer: Question 9. (5 + 6i )(−4 + 7i ) Answer: Question 10. Solve 7x2 + 21 = 0. Answer: Question 11. Find the zeros of f(x) = 2x2 + 32. Answer: 3.3 Completing the Square (pp. 111–118) Question 12. An employee at a local stadium is launching T-shirts from a T-shirt cannon into the crowd during an intermission of a football game. The height h (in feet) of the T-shirt after t seconds can be modeled by h = −16t2 + 96t + 4. Find the maximum height of the T-shirt. Answer: Solve the equation by completing the square. Question 13. x2 + 16x + 17 = 0 Answer: Question 14. 4x2 + 16x + 25 = 0 Answer: Question 15. 9x(x − 6) = 81 Answer: Question 16. Write y = x2 − 2x + 20 in vertex form. Then identify the vertex. Answer: 3.4 Using the Quadratic Formula (pp. 121–130) Solve the equation using the Quadratic Formula. Question 17. −x2 + 5x = 2 Answer: Question 18. 2x2 + 5x = 3 Answer: Question 19. 3x2 − 12x + 13 = 0 Answer: Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. Question 20. −x2 − 6x − 9 = 0 Answer: Question 21. x2 − 2x − 9 = 0 Answer: Question 22. x2 + 6x + 5 = 0 Answer: 3.5 Solving Nonlinear Systems (pp. 131–138) Solve the system by any method. Explain your choice of method. Question 23. 2x2 − 2 = y −2x + 2 = y Answer: Question 24. x2 − 6x + 13 = y −y = −2x + 3 Answer: Question 25. x2 + y2 = 4 −15x + 5 = 5y Answer: Question 26. Solve −3x2 + 5x − 1 = 5x2 − 8x − 3 by graphing. Answer: 3.6 Quadratic Inequalities (pp. 139–146) Graph the inequality. Question 27. y > x2 + 8x + 16 Answer: Question 28. y ≥ x2 + 6x + 8 Answer: Question 29. x2 + y ≤ 7x − 12 Answer: Graph the system of quadratic inequalities. Question 30. x2 − 4x + 8 > y −x2 + 4x + 2 ≤ y Answer: Question 31. 2x2 − x ≥ y − 5 0.5x2> y − 2x− 1 Answer: Question 32. −3x2 − 2x ≤ y + 1 −2x2 + x − 5 > −y Answer: Solve the inequality. Question 33. 3x2 + 3x − 60 ≥0 Answer: Question 34. −x2 − 10x < 21 Answer: Question 35. 3x2 + 2 ≤ 5x Answer: Quadratic Equations and Complex Numbers Chapter Test Solve the equation using any method. Provide a reason for your choice. Question 1. 0 = x2 + 2x + 3 Answer: Question 2. 6x = x2 + 7 Answer: Question 3. x2 + 49 = 85 Answer: Question 4. (x + 4)(x − 1) = −x2 + 3x + 4 Answer: Explain how to use the graph to find the number and type of solutions of the quadratic equation. Justify your answer by using the discriminant. Question 5. $$\frac{1}{2}$$x2 + 3x + $$\frac{9}{2}$$ = 0 Answer: Question 6. 4x2 + 16x + 18 = 0 Answer: Question 7. −x2 + $$\frac{1}{2}$$x + $$\frac{3}{2}$$ = 0 Answer: Solve the system of equations or inequalities. Question 8. x2 + 66 = 16x − y 2x − y = 18 Answer: Question 9. y ≥ $$\frac{1}{4}$$x2 − 2 y < −(x + 3)2x − y = 18 + 4 Answer: Question 10. 0 = x2 + y2 − 40 y = x + 4 Answer: Question 11. Write (3 + 4i )(4 − 6i ) as a complex number in standard form. Answer: Question 12. The aspect ratio of a widescreen TV is the ratio of the screen’s width to its height, or 16 : 9. What are the width and the height of a 32-inch widescreen TV? Justify your answer. (Hint: Use the Pythagorean Theorem and the fact that TV sizes refer to the diagonal length of the screen.) Answer: Question 13. The shape of the Gateway Arch in St. Louis, Missouri, can be modeled by y = −0.0063x2 + 4x, where x is the distance (in feet) from the left foot of the arch and y is the height (in feet) of the arch above the ground. For what distances x is the arch more than 200 feet above the ground? Justify your answer. Answer: Question 14. You are playing a game of horseshoes. One of your tosses is modeled in the diagram, where x is the horseshoe’s horizontal position (in feet) and y is the corresponding height (in feet). Find the maximum height of the horseshoe. Then find the distance the horseshoe travels. Justify your answer. Answer: Quadratic Equations and Complex Numbers Cumulative Assessment Question 1. The graph of which inequality is shown? A. y2 > x2 + x – 6 B. y ≥ x2 + x – 6 C. y > x2 – x – 6 D. y ≥ x2 – x – 6 Answer: Question 2. Classify each function by its function family. Then describe the transformation of the parent function. a. g(x) = x + 4 b. h(x) = 5 c. h(x) = x2 − 7 d. g(x) = −∣x + 3∣− 9 e. g(x) = $$\frac{1}{4}$$(x − 2)2 − 1 f. h(x) = 6x+ 11 Answer: Question 3. Two baseball players hit back-to-back home runs. The path of each home run is modeled by the parabolas below, where x is the horizontal distance (in feet) from home plate and y is the height (in feet) above the ground. Choose the correct symbol for each inequality to model the possible locations of the top of the outfield wall.(HSA-CED.A.3) Answer: Question 4. You claim it is possible to make a function from the given values that has an axis of symmetry at x = 2. Your friend claims it is possible to make a function that has an axis of symmetry at x = −2. What values can you use to support your claim? What values support your friend’s claim?(HSF-IF.B.4) Answer: Question 5. Which of the following values are x-coordinates of the solutions of the system? y = x2 – 6x + 14 y = 2x + 7 Answer: Question 6. The table shows the altitudes of a hang glider that descends at a constant rate. How long will it take for the hang glider to descend to an altitude of 100 feet? Justify your answer. A. 25 seconds B. 35 seconds C. 45 seconds D. 55 seconds Answer: Question 7. Use the numbers and symbols to write the expression x2 + 16 as the product of two binomials. Some may be used more than once. Justify your answer. Answer: Question 8. Choose values for the constants h and k in the equation x = $$\frac{1}{4}$$( y − k)2 + h so that each statement is true.(HSA-CED.A.2) Answer: Question 9. Which of the following graphs represents a perfect square trinomial? Write each function in vertex form by completing the square. Answer: Big Ideas Math Geometry Answers Chapter 8 Similarity Studying & Practicing Math Geometry would be done in a fun learning process for a better understanding of the concepts. So, the best guide to prepare math in a fun learning way is our provided Big Ideas Math Geometry Answers Chapter 8 Similarity Guide. In this study guide, you will discover various exercise questions, chapter reviews, tests, chapter practices, cumulative assessment, etc. to learn all topics of chapter 8 similarity. These questions and answers are explained by the subject experts in a simple manner to make students learn so easily & score maximum marks in the exams. Big Ideas Math Book Geometry Answer Key Chapter 8 Similarity BIM Geometry Book Solutions are available for all chapters along with Chapter 8 Similarity on our website. So, make sure to check all the chapters of Big Ideas Math Book Geometry Answer Key and learn the subject thoroughly. Based on the common core 2019 curriculum, these Big Ideas Math Geometry Answers Chapter 8 Similarity are prepared. So, students can instantly take homework help from BIM Geometry Ch 8 Similarity Answers. Simply tap on the below direct links and refer to the solutions covered in the Big Ideas Math Book Geometry Answer Key Chapter 8 Similarity Guide. Similarity Maintaining Mathematical Proficiency Tell whether the ratios form a proportion. Question 1. $$\frac{5}{3}, \frac{35}{21}$$ Answer: Yes, the ratios $$\frac{5}{3}, \frac{35}{21}$$ form a proportion. Explanation: A proportion means two ratios are equal. So, cross product of $$\frac{5}{3}, \frac{35}{21}$$ is 21 x 5 = 105 = 35 x 3 Therefore, $$\frac{5}{3}, \frac{35}{21}$$ form a proportion. Question 2. $$\frac{9}{24}, \frac{24}{64}$$ Answer: Yes, the ratios $$\frac{9}{24}, \frac{24}{64}$$ form a proportion. Explanation: If the cross product of two ratios is equal, then it forms a proportion. So, 24 x 24 = 576 = 64 x 9 Therefore, the ratios $$\frac{9}{24}, \frac{24}{64}$$ form a proportion. Question 3. $$\frac{8}{56}, \frac{6}{28}$$ Answer: The ratios $$\frac{8}{56}, \frac{6}{28}$$ do not form a proportion. Explanation: If the cross product of two ratios is equal, then it forms a proportion. So, 56 x 6 = 336, 28 x 8 = 224 Therefore, the ratios $$\frac{8}{56}, \frac{6}{28}$$ do not form a proportion. Question 4. $$\frac{18}{4}, \frac{27}{9}$$ Answer: The ratios $$\frac{18}{4}, \frac{27}{9}$$ do not form a proportion. Explanation: If the cross product of two ratios is equal, then it forms a proportion. So, 9 x 18 = 162, 27 x 4 = 108 Therefore, the ratios $$\frac{18}{4}, \frac{27}{9}$$ do not form a proportion. Question 5. $$\frac{15}{21}, \frac{55}{77}$$ Answer: The ratios $$\frac{15}{21}, \frac{55}{77}$$ form a proportion. Explanation: If the cross product of two ratios is equal, then it forms a proportion. So, 15 x 77 = 1155, 55 x 21 = 1155 Therefore, the ratios $$\frac{15}{21}, \frac{55}{77}$$ form a proportion. Question 6. $$\frac{26}{8}, \frac{39}{12}$$ Answer: The ratios $$\frac{26}{8}, \frac{39}{12}$$ form a proportion. Explanation: If the cross product of two ratios is equal, then it forms a proportion. So, 26 x 12 = 312, 39 x 8 = 312 Therefore, the ratios $$\frac{26}{8}, \frac{39}{12}$$ form a proportion. Find the scale factor of the dilation. Question 7. Answer: k = $$\frac { 3 }{ 7 }$$ Explanation: The scale factor k = $$\frac { CP’ }{ CP }$$ = $$\frac { 6 }{ 14 }$$ = $$\frac { 3 }{ 7 }$$ Question 8. Answer: k = $$\frac { 3 }{ 8 }$$ Explanation: The scale factor k = $$\frac { CP }{ CP’ }$$ = $$\frac { 9 }{ 24 }$$ = $$\frac { 3 }{ 8 }$$ Question 9. Answer: k = $$\frac { 1 }{ 2 }$$ Explanation: The scale factor k = $$\frac { MK }{ M’K’ }$$ = $$\frac { 14 }{ 28 }$$ = $$\frac { 1 }{ 2 }$$ Question 10. ABSTRACT REASONING If ratio X and ratio Y form a proportion and ratio Y and ratio Z form a proportion, do ratio X and ratio Z form a proportion? Explain our reasoning. Answer: Yes, ratio X and ratio Z form a proportion. Explanation: If ratios are proportional means they are equal. So, ratio X and ratio Y form a proportion that means X = Y ratio Y and ratio Z form a proportion that means Y = Z From the above two equations, we can say that X = Z So, ratio X and ratio Z also form a proportion. Similarity Mathematical Practices Monitoring Progress Question 1. Find the perimeter and area of the image when the trapezoid is dilated by a scale factor of (a) 2, (b) 3, and (c) 4. Answer: (a) Perimeter is 32 cm, area is 48 sq cm. (b) Perimeter is 48 cm, the area is 108 sq cm. (c) Perimeter is 64 cm, the area is 192 sq cm. Explanation: The perimeter of trapezoid P = 2 + 5 + 6 + 3 = 16 cm Area of trapezoid A = $$\frac { (2 + 6)3 }{ 2 }$$ = $$\frac { 3(8) }{ 2 }$$ = $$\frac { 24 }{ 2 }$$ = 12 sq cm (a) If scale factor k = 2, then Perimeter = kP = 2 x 16 = 32 Area = k²A = 2² x 12 = 4 x 12 = 48 (b) If scale factor k = 3, then, Perimeter = kP = 3 x 16 = 48 cm Area = k²A = 3² x 12 = 9 x 12 = 108 (c) If scale factor k = 4, then Perimeter = kP = 4 x 16 = 64 Area = k²A = 4² x 12 = 16 x 12 = 192 Question 2. Find the perimeter and area of the image when the parallelogram is dilated by a scale factor of (a) 2, (b) 3, and (c) $$\frac{1}{2}$$ Answer: (a) Perimeter is 28 ft, area is 32 sq ft (b) Perimeter is 42 ft, area is 72 sq ft (c) Perimeter is 7 ft, area is 2 sq ft Explanation: Perimeter of parallelogram P = 2(2 + 5) = 7 x 2 = 14 ft Area of the parallelogram = 2 x 4 = 8 sq ft (a) If scale factor k = 2, then Perimeter = kP = 2 x 14 = 28 Area = k²A = 2² x 8 = 4 x 8 = 32 (b) If scale factor k = 3, then Perimeter = kP = 3 x 14 = 42 Area = k²A = 3² x 8 = 72 (c) If scale factor k = $$\frac{1}{2}$$, then Perimeter = kP = $$\frac{1}{2}$$ x 14 = 7 Area = k²A = $$\frac{1}{2²}$$ x 8 = $$\frac{1}{4}$$ x 8 = 2 Question 3. A rectangular prism is 3 inches wide, 4 inches long, and 5 inches tall. Find the surface area and volume of the image of the prism when it is dilated by a scale factor of (a) 2, (b) 3, and (c) 4. Answer: (a) Surface area is 376 sq in, volume is 480 cubic in (b) Surface area is 846 sq in, volume is 1620 cubic in (c) Surface area is 1504 sq in, volume is 3840 cubic in Explanation: The surface area of the rectangular prism A = 2(3 x 4 + 4 x 5 + 5 x 3) = 2(12 + 20 + 15) = 2(47) = 94 in Volume of the rectangular prism V = 3 x 4 x 5 = 60 in³ (a) If the scale factor k = 2, then Surface Area = k²A = 2² x 94 = 4 x 94 = 376 sq in Volume = k³V = 2³ x 60 = 8 x 60 = 480 cubic in (b) If the scale factor k = 3, then Surface Area = k²A = 3² x 94 = 9 x 94 = 846 Volume = k³V = 3³ x 60 = 27 x 60 = 1620 (c) If the scale factor k = 4, then Surface Area = k²A = 4² x 94 = 16 x 94 = 1504 Volume = k³V = 4³ x 60 = 64 x 60 = 3840 8.1 Similar Polygons Exploration 1 Comparing Triangles after a Dilation Work with a partner: Use dynamic geometry software to draw any ∆ABC. Dilate ∆ABC to form a similar ∆A’B’C’ using an scale factor k and an center of dilation. a. Compare the corresponding angles of ∆A’B’C and ∆ABC. Answer: b. Find the ratios of the lengths of the sides of ∆A’B’C’ to the lengths of the corresponding sides of ∆ABC. What do you observe? Answer: c. Repeat parts (a) and (b) for several other triangles, scale factors, and centers of dilation. Do you obtain similar results? Answer: Exploration 2 Comparing Triangles after a Dilation Work with a partner: Use dynamic geometry Software to draw any ∆ABC. Dilate ∆ABC to form a similar ∆A’B’C’ using any scale factor k and any center of dilation. a. Compare the perimeters of ∆A’B’C and ∆ABC. What do you observe? Answer: b. Compare the areas of ∆A’B’C’ and ∆ABC. What do you observe? Answer: c. Repeat parts (a) and (b) for several other triangles, scale factors, and centers of dilation. Do you obtain similar results? LOOKING FOR STRUCTURE To be proficient in math, you need to look closely to discern a pattern or structure. Answer: Communicate Your Answer Question 3. How are similar polygons related? Answer: if two polygons are similar means they have the same shape, corresponding angles are congruent and the ratios of lengths of their corresponding sides are equal. The common ratio is called the scale factor. Question 4. A ∆RST is dilated by a scale factor of 3 to form ∆R’S’T’. The area of ∆RST is 1 square inch. What is the area of ∆R’S’T’? Answer: Area of ∆R’S’T’ = 9 sq in Explanation: Given that, Area of ∆RST = 1 sq inch Scale factor k = 3 Area of ∆R’S’T’ = k² x Area of ∆RST = 3² x 1 = 9 x 1 = 9 Lesson 8.1 Similar Polygons Monitoring Progress Question 1. In the diagram, ∆JKL ~ ∆PQR. Find the scale factor from ∆JKL to ∆PQR. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality. Answer: The pairs of congruent angles are ∠K = ∠Q, ∠J = ∠P, ∠ L = ∠R The scale factor is $$\frac { 3 }{ 2 }$$ The ratios of the corresponding side lengths in a statement of proportionality are $$\frac { PQ }{ JK } = \frac { PR }{ JL } = \frac { QR }{ LK }$$ Explanation: Given that, ∆JKL ~ ∆PQR The pairs of congruent angles are ∠K = ∠Q, ∠J = ∠P, ∠ L = ∠R To find the scale factor, $$\frac { PQ }{ JK } = \frac { 9 }{ 6 }$$ = $$\frac { 3 }{ 2 }$$, $$\frac { PR }{ JL } = \frac { 12 }{ 8 }$$ = $$\frac { 3 }{ 2 }$$, $$\frac { QR }{ LK } = \frac { 6 }{ 4 }$$ = $$\frac { 3 }{ 2 }$$ So, the scale factor is $$\frac { 3 }{ 2 }$$ Question 2. Find the value of x. ABCD ~ QRST Answer: x = 2 Explanation: The triangles are similar, so corresponding side lengths are proportional. $$\frac { RS }{ BC }$$ = $$\frac { AB }{ QR }$$ $$\frac { 4 }{ x }$$ = $$\frac { 12 }{ 6 }$$ $$\frac { 4 }{ x }$$ = 2 4 = 2x x = 2 Question 3. Find KM ∆JKL ~ ∆EFG Answer: KM = 42 Explanation: Scale factor = $$\frac { JM }{ GH }$$ = $$\frac { 48 }{ 40 }$$ = $$\frac { 6 }{ 5 }$$ Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion $$\frac { KM }{ HF }$$ = $$\frac { 6 }{ 5 }$$ $$\frac { KM }{ 35 }$$ = $$\frac { 6 }{ 5 }$$ KM = $$\frac { 6 }{ 5 }$$ x 35 KM = 42 Question 4. The two gazebos shown are similar pentagons. Find the perimeter of Gazebo A. Answer: Perimeter of Gazebo A = 46 m Explanation: Scale factor = $$\frac { AB }{ FG }$$ = $$\frac { 10 }{ 15 }$$ = $$\frac { 2 }{ 3 }$$ So, $$\frac { AE }{ FK }$$ = $$\frac { 2 }{ 3 }$$ $$\frac { x }{ 18 }$$ = $$\frac { 2 }{ 3 }$$ x = 12 $$\frac { ED }{ KJ }$$ = $$\frac { 2 }{ 3 }$$ $$\frac { ED }{ 15 }$$ = $$\frac { 2 }{ 3 }$$ ED = 10 $$\frac { DC }{ JH }$$ = $$\frac { 2 }{ 3 }$$ $$\frac { DC }{ 12 }$$ = $$\frac { 2 }{ 3 }$$ DC = 8 $$\frac { BC }{ GH }$$ = $$\frac { 2 }{ 3 }$$ $$\frac { BC }{ 9 }$$ = $$\frac { 2 }{ 3 }$$ BC = 6 Therefore, perimeter = 6 + 8 + 10 + 12 + 10 = 46 Question 5. In the diagram, GHJK ~ LMNP. Find the area of LMNP. Area of GHJK = 84m2 Answer: Area of LMNP = 756 m2 Explanation: As shapes are similar, their corresponding side lengths are proportional. Scale Factor k = $$\frac { NP }{ JK }$$ = $$\frac { 21 }{ 7 }$$ = 3 Area of LMNP = k² x Area of GHJK = 3² x 84 = 756 m2 Question 6. Decide whether the hexagons in Tile Design 1 are similar. Explain. Answer: Question 7. Decide whether the hexagons in Tile Design 2 are similar. Explain. Answer: Exercise 8.1 Similar Polygons Vocabulary and Core Concept Check Question 1. COMPLETE THE SENTENCE For two figures to be similar, the corresponding angles must be ____________ . and the corresponding side lengths must be _____________ . Answer: Question 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. What is the scale factor? Answer: Scale Factor = $$\frac{image-length}{actual-length}$$ = $$\frac{3}{12}$$= $$\frac{4}{16}$$= $$\frac{5}{20}$$ = $$\frac{1}{4}$$ What is the ratio of their areas? Answer: What is the ratio of their corresponding side lengths? Answer: What is the ratio of their perimeters? Answer: Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, find the scale factor. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality. Question 3. ∆ABC ~ ∆LMN Answer: Question 4. DEFG ~ PQRS Answer: In Exercises 5-8, the polygons are similar. Find the value of x. Question 5. Answer: Question 6. Answer: x = 20 Explanation: $$\frac { DF }{ GJ }$$ = $$\frac { DE }{ GH }$$ $$\frac { 16 }{ 12 }$$ = $$\frac { x }{ 15 }$$ x = $$\frac { 16 x 15 }{ 12 }$$ = $$\frac { 240 }{ 12 }$$ x = 20 Question 7. Answer: Question 8. Answer: x = 12 Explanation: $$\frac { PN }{ KJ }$$ = $$\frac { MN }{ JH }$$ $$\frac { x }{ 8 }$$ = $$\frac { 9 }{ 6 }$$ x = $$\frac { 9 x 8 }{ 6 }$$ = $$\frac { 72 }{ 6 }$$ x = 12 In Exercises 9 and 10, the black triangles are similar. Identify the type of segment shown in blue and find the value of the variab1e. Question 9. Answer: Question 10. Answer: Explanation: $$\frac { y }{ 18 }$$ = $$\frac { y – 1 }{ 16 }$$ 18(y – 1) = 16y 18y – 18 = 16y 18y – 16y = 18 2y = 18 y = 9 In Exercises 11 and 12, RSTU ~ ABCD. Find the ratio of their perimeters. Question 11. Answer: Question 12. Answer: $$\frac { RS + ST + TU + UR }{ AB + BC + CD + DA }$$ = $$\frac { RS }{ AB }$$ = $$\frac { 18 }{ 24 }$$ The ratio of perimeter is $$\frac { 3 }{ 4 }$$. In Exercises 13-16, two polygons are similar. The perimeter of one polygon and the ratio of the corresponding side lengths are given. Find the perimeter of the other polygon. Question 13. perimeter of smaller polygon: 48 cm: ratio: $$\frac{2}{3}$$ Answer: Question 14. perimeter of smaller polygon: 66 ft: ratio: $$\frac{3}{4}$$ Answer: The perimeter of larger polygon is 88 ft. Explanation: $$\frac { smaller }{ larger }$$ = $$\frac { 66 }{ x }$$ = $$\frac{3}{4}$$ 66 x 4 = 3x 3x = 264 x = $$\frac { 264 }{ 3 }$$ = 88 Question 15. perimeter of larger polygon: 120 yd: rttio: $$\frac{1}{6}$$ Answer: Question 16. perimeter of larger polygon: 85 m; ratio: $$\frac{2}{5}$$ Answer: The perimeter of smaller polygon is 34 m. Explanation: $$\frac { smaller }{ larger }$$ = $$\frac { x }{ 85 }$$ = $$\frac{2}{5}$$ 85 x 2 = 5x 5x = 170 x = $$\frac { 170 }{ 5 }$$ = 34 Question 17. MODELING WITH MATHEMATICS A school gymnasium is being remodeled. The basketball court will be similar to an NCAA basketball court, which has a length of 94 feet and a width of 50 feet. The school plans to make the width of the new court 45 feet. Find the perimeters of ail NCAA court and of the new court in the school. Answer: Question 18. MODELING WITH MATHEMATICS Your family has decided to put a rectangular patio in your backyard. similar to the shape of your backyard. Your backyard has a length of 45 feet and a width of 20 feet. The length of your new patio is 18 feet. Find the perimeters of your backyard and of the patio. Answer: The perimeter of the backyard is 130 ft. Perimeter of patio is 52 ft Explanation: Draw a rectangle to represent the patio and a larger rectangle to represent our backyard and its going to similar figures Perimeter of backyard = 2(45 + 20) = 2(65) = 130 ft Scale factor = $$\frac { 18 }{ 45 }$$ = $$\frac { 2 }{ 5 }$$ So, $$\frac { perimeter of patio }{ perimeter of backyard }$$ = $$\frac { 2 }{ 5 }$$ $$\frac { perimeter of patio }{ 130 }$$ = $$\frac { 2 }{ 5 }$$ Perimeter of patio = $$\frac { 260 }{ 5 }$$ = 52 ft In Exercises 19-22, the polygons are similar. The area of one polygon is given. Find the area of the other polygon. Question 19. Answer: Question 20. Answer: Area of the larger triangle is 90 cm² Explanation: $$\frac { 10 }{ A }$$ = ($$\frac { 4 }{ 12 }$$)² $$\frac { 10 }{ A }$$ = $$\frac { 1 }{ 9 }$$ A = 10 x 9 A = 90 Question 21. Answer: Question 22. Answer: Area of smaller triangle = 6 sq cm Explanation: $$\frac { A }{ 96 }$$ = ($$\frac { 3 }{ 12 }$$)² $$\frac { A }{ 96 }$$ = $$\frac { 1 }{ 16 }$$ 16A = 96 A = $$\frac { 96 }{ 16 }$$ A = 6 Question 23. ERROR ANALYSIS Describe and correct the error in finding the perimeter of triangle B. The triangles are similar. Answer: Question 24. ERROR ANALYSIS Describe and correct the error in finding the area of triangle B. The triangles are similar. Answer: Because the first ratio has a side of A over the side length of B, the square of the second ratio should have the area of B over the area of A. $$\frac { 24 }{ x }$$ = ($$\frac { 6 }{ 18 }$$)² $$\frac { 24 }{ x }$$ = $$\frac { 1 }{ 9 }$$ x = 24 x 9 x = 216 In Exercises 25 and 26, decide whether the red and blue polygons are similar. Question 25. Answer: Question 26. Answer: Yes Because both shapes are apparent and their side lengths are proportional and their corresponding angles are congruent. Question 27. REASONING Triangles ABC and DEF are similar. Which statement is correct? Select all that apply. (A) $$\frac{B C}{E F}=\frac{A C}{D F}$$ (B) $$\frac{A B}{D E}=\frac{C A}{F E}$$ (C) $$\frac{A B}{E F}=\frac{B C}{D E}$$ (D) $$\frac{C A}{F D}=\frac{B C}{E F}$$ Answer: ANALYZING RELATIONSHIPS In Exercises 28 – 34, JKLM ~ EFGH. Question 28. Find the scale factor of JKLM to EFGH. Answer: scale factor = $$\frac { EF }{ JK }$$ =$$\frac { 8 }{ 20 }$$ k = $$\frac { 2 }{ 5 }$$ Question 29. Find the scale factor of EFGH to JKLM. Answer: Question 30. Find the values of x, y, and z. Answer: x = $$\frac { 55 }{ 2 }$$ y = 12 z = 65° Explanation: $$\frac { KL }{ GF }$$ = $$\frac { x }{ 11 }$$ = $$\frac { 5 }{ 2 }$$ 2x = 55 x = $$\frac { 55 }{ 2 }$$ $$\frac { MJ }{ HE }$$ = $$\frac { 30 }{ y }$$ = $$\frac { 5 }{ 2 }$$ 5y = 60 y = 12 Question 31. Find the perimeter of each polygon. Answer: Question 32. Find the ratio of the perimeters of JKLM to EFGH. Answer: The perimeter of JKLM : Perimeter of EFGH = 85 : 34 Question 33. Find the area of each polygon. Answer: Question 34. Find the ratio of the areas of JKLM to EFGH. Answer: Area of JKLM : Area of EFGH = 378.125 : 60.5 = 25 : 4 Question 35. USING STRUCTURE Rectangle A is similar to rectangle B. Rectangle A has side lengths of 6 and 12. Rectangle B has a side length of 18. What are the possible values for the length of the other side of rectangle B? Select all that apply. (A) 6 (B) 9 (C) 24 (D) 36 Answer: Question 36. DRAWING CONCLUSIONS In table tennis, the table is a rectangle 9 feet long and 5 feet wide. A tennis Court is a rectangle 78 feet long and 36 feet wide. Are the two surfaces similar? Explain. If so, find the scale factor of the tennis court to the table. Answer: The tennis table and court are not similar Explanation: If two figures are similar then their angles are congruent and sides are proportional. If the tennis court and table are similar, then $$\frac { length of table }{ length of court }$$ = $$\frac { width of table }{ width of court }$$ $$\frac { 9 }{ 78 }$$ = $$\frac { 5 }{ 36 }$$ 9 • 36 = 5 • 78 324 = 390 So, Table and court are not similar. MATHEMATICAL CONNECTIONS In Exercises 37 and 38, the two polygons are similar. Find the values of x and y. Question 37. Answer: Question 38. Answer: x = 7.5 Explanation: $$\frac { x }{ 5 }$$ = $$\frac { 6 }{ 4 }$$ x = $$\frac { 15 }{ 2 }$$ ATTENDING TO PRECISION In Exercises 39 – 42. the figures are similar. Find the missing corresponding side length. Question 39. Figure A has a pen meter of 72 meters and one of the side lengths is 18 meters. Figure B has a perimeter of 120 meters. Answer: Question 40. Figure A has a perimeter of 24 inches. Figure B has a perimeter of 36 inches and one of the side lengths is 12 inches. Answer: The corresponding side length of figure A is 8 in Explanation: $$\frac { Perimeter of A }{ Perimeter of B }$$ = $$\frac { Side length of A }{ Side length of B }$$ $$\frac { 24 }{ 36 }$$ = $$\frac { x }{ 12 }$$ $$\frac { 2 }{ 3 }$$ = $$\frac { x }{ 12 }$$ 12 • 2 = 3x x = 8 Question 41. Figure A has an area of 48 square feet and one of the side lengths is 6 feet. Figure B has an area of 75 square feet. Answer: Question 42. Figure A has an area of 18 square feet. Figure B has an area of 98 square feet and one of the side lengths is 14 feet. Answer: The corresponding side length of figure A is 6 ft. Explanation: $$\frac { Area of A }{ Area of B }$$ = ($$\frac { Side length of A }{ Side length of B }$$)² $$\frac { 18 }{ 98 }$$ = ($$\frac { x }{ 14 }$$)² $$\frac { 9 }{ 49 }$$ = $$\frac { x² }{ 196 }$$ x² = 36 x = 6 CRITICAL THINKING In Exercises 43-48, tell whether the polygons are always, sometimes, or never similar. Question 43. two isosceles triangles Answer: Question 44. two isosceles trapezoids Answer: Two isosceles trapezoids are sometimes similar. Question 45. two rhombuses Answer: Question 46. two squares Answer: Two squares are always similar. Question 47. two regular polygons Answer: Question 48. a right triangle and an equilateral triangle Answer: A right triangle and an equilateral triangle are never similar. Question 49. MAKING AN ARGUMENT Your sister claims that when the side lengths of two rectangles are proportional, the two rectangles must be similar. Is she correct? Explain your reasoning. Answer: Question 50. HOW DO YOU SEE IT? You shine a flashlight directly on an object to project its image onto a parallel screen. Will the object and the image be similar? Explain your reasoning. Answer: The object and image are similar. Question 51. MODELING WITH MATHEMATICS During a total eclipse of the Sun, the moon is directly in line with the Sun and blocks the Sun’s rays. The distance DA between Earth and the Sun is 93,00,000 miles. the distance DE between Earth and the moon is 2,40,000 miles, and the radius AB of the Sun is 432,5000 miles. Use the diagram and the given measurements to estimate the radius EC of the moon. Answer: Question 52. PROVING A THEOREM Prove the Perimeters of Similar Polygons Theorem (Theorem 8.1) for similar rectangles. Include a diagram in your proof. Answer: $$\frac { PQ + QR + RS + SP }{ KL + LM + MN + NK }$$ = $$\frac { PQ }{ KL }$$ = $$\frac { QR }{ LM }$$ = $$\frac { RS }{ MN }$$ = $$\frac { SP }{ NK }$$ Question 53. PROVING A THEOREM Prove the Areas of Similar Polygons Theorem (Theorem 8.2) for similar rectangles. Include a diagram in our proof. Answer: Question 54. THOUGHT PROVOKING The postulates and theorems in this book represent Euclidean geometry. In spherical geometry. all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. A plane is the surface of the sphere. In spherical geometry, is it possible that two triangles are similar but not congruent? Explain your reasoning. Answer: Question 55. CRITICAL THINKING In the diagram, PQRS is a square, and PLMS ~ LMRQ. Find the exact value of x. This value is called the golden ratio. Golden rectangles have their length and width in this ratio. Show that the similar rectangles in the diagram are golden rectangles. Answer: Question 56. MATHEMATICAL CONNECTIONS The equations of the lines shown are y = $$\frac{4}{3}$$x + 4 and y = $$\frac{4}{3}$$x – 8. Show that ∆AOB ~ ∆COD. Answer: The two lines slopes are equal and triangles angles are congruent and side lengths are proportional. So, triangles are similar. Maintaining Mathematical proficiency Find the value of x. Question 57. Answer: Question 58. Answer: x = 66° Explanation: x + 24 + 90 = 180 x + 114 = 180 x = 180 – 114 x = 66 Question 59. Answer: Question 60. Answer: x = 60° Explanation: x + x + x = 180 3x = 180 x = 60 8.2 Proving Triangle Similarity by AA Exploration 1 Comparing Triangles Work with a partner. Use dynamic geometry software. a. Construct ∆ABC and ∆DEF So that m∠A = m∠D = 106°, m∠B = m∠E = 31°, and ∆DEF is not congruent to ∆ABC. Answer: m∠C ≠ m∠F b. Find the third angle measure and the side lengths of each triangle. Copy the table below and record our results in column 1. Answer: c. Are the two triangles similar? Explain. CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results in constructing arguments. Answer: d. Repeat parts (a) – (c) to complete columns 2 and 3 of the table for the given angle measures. Answer: e. Complete each remaining column of the table using your own choice of two pairs of equal corresponding angle measures. Can you construct two triangles in this way that are not similar? Answer: f. Make a conjecture about any two triangles with two pairs of congruent corresponding angles. Answer: Communicate Your Answer Question 2. What can you conclude about two triangles when you know that two pairs of corresponding angles are congruent? Answer: Question 3. Find RS in the figure at the left. Answer: Lesson 8.2 Proving Triangle Similarity by AA Monitoring Progress Show that the triangles are similar. Write a similarity statement. Question 1. ∆FGH and ∆RQS Answer: ∆FGH and ∆RQS are similar by the AA similarity theorem. Question 2. ∆CDF and ∆DEF Answer: Question 3. WHAT IF? Suppose that $$\overline{S R}$$ $$\overline{T U}$$ in Example 2 part (b). Could the triangles still be similar? Explain. Answer: Question 4. WHAT IF? A child who is 58 inches tall is standing next to the woman in Example 3. How long is the child’s shadow’? Answer: Question 5. You are standing outside, and you measure the lengths 0f the shadows cast by both you and a tree. Write a proportion showing how you could find the height of the tree. Answer: Exercise 8.2 Proving Triangle Similarity by AA Vocabulary and Core Concept Check Question 1. COMPLETE THE SENTENCE If two angles of one triangle are congruent to two angles of another triangle. then the triangles are __________ . Answer: Question 2. WRITING Can you assume that corresponding sides and corresponding angles of any two similar triangles are congruent? Explain. Answer: Monitoring Progress and Modeling with Mathematics In Exercises 3 – 6. determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. Question 3. Answer: Question 4. Answer: Question 5. Answer: Question 6. Answer: In Exercises 7 – 10. show that the two triangles are similar. Question 7. Answer: Question 8. Answer: Question 9. Answer: Question 10. Answer: In Exercises 11 – 18, use the diagram to copy and complete the statement. Question 11. ∆CAG ~ _________ Answer: Question 12. ∆DCF ~ _________ Answer: Question 13. ∆ACB ~ _________ Answer: Question 14. m∠ECF = _________ Answer: Question 15. m∠ECD = _________ Answer: Question 16. CF = _________ Answer: Question 17. BC = _________ Answer: Question 18. DE = _________ Answer: Question 19. ERROR ANALYSIS Describe and correct the error in using the AA Similarity Theorem (Theorem 8.3). Answer: Question 20. ERROR ANALYSIS Describe and correct the error in finding the value of x. Answer: Question 21. MODELING WITH MATHEMATICS You can measure the width of the lake using a surveying technique, as shown in the diagram. Find the width of the lake, WX. Justify your answer. Answer: Question 22. MAKING AN ARGUMENT You and your cousin are trying to determine the height of a telephone pole. Your cousin tells you to stand in the pole’s shadow so that the tip of your shadow coincides with the tip of the pole’s shadow. Your Cousin claims to be able to use the distance between the tips of the shadows and you, the distance between you and the pole, and your height to estimate the height of the telephone pole. Is this possible? Explain. Include a diagram in your answer. Answer: REASONING In Exercises 23 – 26, is it possible for ∆JKL and ∆XYZ to be similar? Explain your reasoning. Question 23. m∠J = 71°, m∠K = 52°, m∠X = 71°, and m∠Z = 57° Answer: Question 24. ∆JKL is a right triangle and m∠X + m∠Y= 150°. Answer: Question 25. m∠L = 87° and m∠Y = 94° Answer: Question 26. m∠J + m∠K = 85° and m∠Y + m∠Z = 80° Answer: Question 27. MATHEMATICAL CONNECTIONS Explain how you can use similar triangles to show that any two points on a line can be used to find its slope. Answer: Question 28. HOW DO YOU SEE IT? In the diagram, which triangles would you use to find the distance x between the shoreline and the buoy? Explain your reasoning. Answer: Question 29. WRITING Explain why all equilateral triangles are similar. Answer: Question 30. THOUGHT PROVOKING Decide whether each is a valid method of showing that two quadrilaterals are similar. Justify your answer. a. AAA Answer: b. AAAA Answer: Question 31. PROOF Without using corresponding lengths in similar polygons. prove that the ratio of two corresponding angle bisectors in similar triangles is equal to the scale factor. Answer: Question 32. PROOF Prove that if the lengths of two sides of a triangle are a and b, respectively, then the lengths of the corresponding altitudes to those sides are in the ratio $$\frac{b}{a}$$. Answer: Question 33. MODELING WITH MATHEMATICS A portion of an amusement park ride is shown. Find EF. Justify your answer. Answer: Maintaining Mathematical Practices Determine whether there is enough information to prove that the triangles are congruent. Explain your reasoning. Question 34. Answer: Question 35. Answer: Question 36. Answer: 8.1 & 8.2 Quiz List all pairs of congruent angles. Then write the ratios of the corresponding side lengths in a statement of proportionality. Question 1. ∆BDG ~ ∆MPQ Answer: Question 2. DEFG ~ HJKL Answer: The polygons are similar. Find the value of x. Question 3. Answer: Question 4. Answer: Determine whether the polygons are similar. If they are, write a similarity statement. Explain your reasoning. (Section 8.1 and Section 8.2) Question 5. Answer: Question 6. Answer: Question 7. Answer: Show that the two triangles are similar. Question 8. Answer: Question 9. Answer: Question 10. Answer: Question 11. The dimensions of an official hockey rink used by the National Hockey League (NHL) are 200 feet by 85 feet. The dimensions of an air hockey table are 96 inches by 408 inches. Assume corresponding angles are congruent. (Section 8.1) a. Determine whether the two surfaces are similar. Answer: b. If the surfaces are similar, find the ratio of their perimeters and the ratio ol their areas. If not, find the dimensions of an air hockey table that are similar to an NHL hockey rink. Answer: Question 12. you and a friend buy camping tents made by the same company but in different sizes and colors. Use the information given in the diagram to decide whether the triangular faces of the tents are similar. Explain your reasoning. (Section 8.2) Answer: 8.3 Proving Triangle Similarity by SSS and SAS Exploration 1 Deciding Whether Triangles Are Similar Work with a partner: Use dynamic geometry software. a. Construct ∆ABC and ∆DEF with the side lengths given in column 1 of the table below. Answer: b. Copy the table and complete column 1. Answer: c. Are the triangles similar? Explain your reasoning. Answer: d. Repeat parts (a) – (c) for columns 2 – 6 in the table. Answer: e. How are the corresponding side lengths related in each pair of triangles that are similar? Is this true for each pair of triangles that are not similar? Answer: f. Make a conjecture about the similarity of two triangles based on their corresponding side lengths. CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to analyze situations by breaking them into cases and recognize and use counter examples. Answer: g. Use your conjecture to write another set of side lengths of two similar triangles. Use the side lengths to complete column 7 of the table. Answer: Exploration 2 Deciding Whether Triangles Are Similar Work with a partner: Use dynamic geometry software. Construct any ∆ABC. a. Find AB, AC, and m∠A. Choose any positive rational number k and construct ∆DEF so that DE = k • AB, DF = k • AC, and m∠D = m∠A. Answer: b. Is ∆DEF similar to ∆ABC? Explain your reasoning. Answer: c. Repeat parts (a) and (b) several times by changing ∆ABC and k. Describe your results. Answer: Communicate Your Answer Question 3. What are two ways to use corresponding sides of two triangles to determine that the triangles are similar? Answer: Lesson 8.3 Proving Triangle Similarity by SSS and SAS Monitoring progress Use the diagram. Question 1. Which of the three triangles are similar? Write a similarity statement. Answer: The ratios are equal. So, △LMN, △XYZ are similar. The ratios are not equal. So △LMN, △RST are not similar. Explanation: Compare △LMN, △XYZ by finding the ratios of corresponding side lengths Shortest sides: $$\frac { LM }{ YZ }$$ = $$\frac { 20 }{ 30 }$$ = $$\frac { 2 }{ 3 }$$ Longest sides: $$\frac { LN }{ XY }$$ = $$\frac { 26 }{ 39 }$$ = $$\frac { 2 }{ 3 }$$ Remaining sides: $$\frac { MN }{ ZX }$$ = $$\frac { 24 }{ 36 }$$ = $$\frac { 2 }{ 3 }$$ The ratios are equal. So, △LMN, △XYZ are similar. Compare △LMN, △RST by finding the ratios of corresponding side lengths Shortest sides: $$\frac { LM }{ RS }$$ = $$\frac { 20 }{ 24 }$$ = $$\frac { 5 }{ 6 }$$ Longest sides: $$\frac { LN }{ ST }$$ = $$\frac { 26 }{ 33 }$$ Remaining sides: $$\frac { MN }{ RT }$$ = $$\frac { 24 }{ 30 }$$ = $$\frac { 4 }{ 5 }$$ The ratios are not equal. So △LMN, △RST are not similar. Question 2. The shortest side of a triangle similar to ∆RST is 12 units long. Find the other side 1enths of the triangle. Answer: The other side lengths of the triangle are 15 units, 16.5 units. Explanation: The shortest side of a triangle similar to ∆RST is 12 units Scale factor = $$\frac { 12 }{ 24 }$$ = $$\frac { 1 }{ 2 }$$ So, other sides are 33 x $$\frac { 12 }{ 2 }$$ = 16.5, 30 x $$\frac { 12 }{ 2 }$$ = 15. Explain how to show that the indicated triangles are similar. Question 3. ∆SRT ~ ∆PNQ Answer: The shorter sides: $$\frac { 18 }{ 24 }$$ = $$\frac { 3 }{ 4 }$$ Longer sides: $$\frac { 21 }{ 28 }$$ = $$\frac { 3 }{ 4 }$$ The side lengths are proportional. So ∆SRT ~ ∆PNQ Question 4. ∆XZW ~ ∆YZX Answer: ∆XZW and ∆YZX are not proportional. Explanation: The shorter sides: $$\frac { 9 }{ 16 }$$ Longer sides: $$\frac { 15 }{ 20 }$$ = $$\frac { 3 }{ 4 }$$ The side lengths are not proportional. So ∆XZW and ∆YZX are not proportional. Exercise 8.3 Proving Triangle Similarity by SSS and SAS Vocabulary and Core Concept Check Question 1. COMPLETE THE SENTENCE You plan to show that ∆QRS is similar to ∆XYZ by the SSS Similarity Theorem (Theorem 8.4). Copy and complete the proportion that you will use: Answer: Question 2. WHICH ONE DOESN’T BELONG? Which triangle does not belong with the other three? Explain your reasoning. Answer: Monitoring progress and Modeling with Mathematics In Exercises 3 and 4, determine whether ∆JKL or ∆RST is similar to ∆ABC. Question 3. Answer: Question 4. Answer: In Exercises 5 and 6, find the value of x that makes ∆DEF ~ ∆XYZ. Question 5. Answer: Question 6. Answer: In Exercises 7 and 8, verify that ∆ABC ~ ∆DEF Find the scale factor of ∆ABC to ∆DEF Question 7. ∆ABC: BC = 18, AB = 15, AC = 12 ∆DEF: EF = 12, DE = 10, DF = 8 Answer: Question 8. ∆ABC: AB = 10, BC = 16, CA = 20 ∆DEF: DE = 25, EF = 40, FD =50 Answer: In Exercises 9 and 10. determine whether the two triangles are similar. If they are similar, write a similarity statement and find the scale factor of triangle B to triangle A. Question 9. Answer: Question 10. Answer: In Exercises 11 and 12, sketch the triangles using the given description. Then determine whether the two triangles can be similar. Question 11. In ∆RST, RS = 20, ST = 32, and m∠S = 16°. In ∆FGH, GH = 30, HF = 48, and m∠H = 24°. Answer: Question 12. The side lengths of ∆ABC are 24, 8x, and 48, and the side lengths of ∆DEF are 15, 25, and 6x. Answer: $$\frac { AB }{ DE }$$ = $$\frac { AC }{ DF }$$ = $$\frac { BC }{ EF }$$ $$\frac { 24 }{ 15 }$$ = $$\frac { 8x }{ 25 }$$ x = 5 In Exercises 13 – 16. show that the triangles are similar and write a similarity statement. Explain your reasoning. Question 13. Answer: Question 14. Answer: Question 15. Answer: Question 16. Answer: In Exercises 17 and 18, use ∆XYZ. Question 17. The shortest side of a triangle similar to ∆XYZ is 20 units long. Find the other side lengths of the triangle. Answer: Question 18. The longest side of a triangle similar to ∆XYZ is 39 units long. Find the other side lengths of the triangle. Answer: Question 19. ERROR ANALYSIS Describe and correct the error in writing a similarity statement. Answer: Question 20. MATHEMATICAL CONNECTIONS Find the value of n that makes ∆DEF ~ ∆XYZ when DE = 4, EF = 5, XY = 4(n + 1), YZ = 7n – 1, and ∠E ≅ ∠Y. Include a sketch. Answer: $$\frac { DE }{ XY }$$ = $$\frac { EF }{ YZ }$$ $$\frac { 4 }{ 4(n + 1) }$$ = $$\frac { 5 }{ 7n – 1 }$$ cross multiply the fractions 4(7n – 1) = 20(n + 1) 28n – 4 = 20n + 20 28n – 20n = 20 + 4 8n = 24 n = $$\frac { 24 }{ 8 }$$ n = 3 ATTENDING TO PRECISION In Exercises 21 – 26, use the diagram to copy and complete the statement. Question 21. m∠LNS = ___________ Answer: Question 22. m∠NRQ = ___________ Answer: m∠NRQ = m∠NRP = 91° by the vertical congruence Question 23. m∠NQR = ___________ Answer: Question 24. RQ = ___________ Answer: RQ = 4√3 Explanation: Using the pythogrean theorem NQ² = NR² + RQ² 8² = 4² + RQ² 64 = 16 + RQ² 64 – 16 = RQ² 48 = RQ² RQ = 4√3 Question 25. m∠NSM = ___________ Answer: Question 26. m∠NPR = ___________ Answer: m∠NPR = 28° Explanation: m∠NPR + m∠NRP + m∠RNP = 180° m∠NPR + 91° + 61° = 180° m∠NPR + 152° = 180° m∠NPR = 180° – 152° m∠NPR = 28° Question 27. MAKING AN ARGUMENT Your friend claims that ∆JKL ~ ∆MNO by the SAS Similarity Theorem (Theorem 8.5) when JK = 18, m∠K = 130° KL = 16, MN = 9, m∠N = 65°, and NO = 8, Do you support your friend’s claim? Explain your reasoning. Answer: Question 28. ANALYZING RELATIONSHIPS Certain sections of stained glass are sold in triangular, beveled pieces. Which of the three beveled pieces, if any, are similar? Answer: Out of three triangles, violet and blue triangles are similar. Explanation: Check the similarity of maroon and violet triangles. longest sides: $$\frac { 5 }{ 7 }$$ shortest sides: $$\frac { 3 }{ 4 }$$ remaining sides: $$\frac { 3 }{ 4 }$$ The ratios are not equal. So those traingles are not similar. Check the similarity of blue and violet triangles. longest sides: $$\frac { 5 }{ 5.25 }$$ = 1 shortest sides: $$\frac { 3 }{ 3 }$$ = 1 remaining sides: $$\frac { 3 }{ 3 }$$ = 1 The ratios are equal. So those traingles are similar. Question 29. ATTENDING TO PRECISION In the diagram, $$\frac{M N}{M R}=\frac{M P}{M Q}$$ Which of the statements must be true? Select all that apply. Explain your reasoning. (A) ∠1 ≅∠2 (B) $$\overline{Q R}$$ || $$\overline{N P}$$ (C)∠1 ≅ ∠4 (D) ∆MNP ~ ∆MRQ Answer: Question 30. WRITING Are any two right triangles similar? Explain. Answer: Yes, any two right triangles can be similar. If two right triangles are similar, then the ratio of their longest, smallest and remaining side lengths must be equal and their angles must be congruent. Question 31. MODELING WITH MATHEMATICS In the portion of the shuffleboard court shown, $$\frac{B C}{A C}=\frac{B D}{A E}$$ a. What additional information do you need to show that ∆BED ~ ∆ACE using the SSS Similarity Theorem (Theorem 8.4)? b. What additional information do, you need to show that ∆BCD ~ ∆ACE using the SAS Similarity Theorem (Theorem 8.5)? Answer: Question 32. PROOF Given that ∆BAC is a right triangle and D, E, and F are midpoints. prove that m∠DEF = 90°. Answer: By observing the triangle ABC, m∠BAC = 90° Join the midpoints of the sides of the triangle. m∠DEF = 90° Question 33. PROVING A THEOREM Write a two-column proof of the SAS Similarity Theorem (Theorem 8.5). Given ∠A ≅ ∠D, $$\frac{A B}{D E}=\frac{A C}{D F}$$ Prove ∆ABC ~ ∆DEF Answer: Question 34. CRITICAL THINKING You are given two right triangles with one pair of corresponding legs and the pair of hypotenuses having the same length ratios. a. The lengths of the given pair of corresponding legs are 6 and 18, and the lengths of the hypotenuses are 10 and 30. Use the Pythagorean Theorem to find the lengths of the other pair of corresponding legs. Draw a diagram. Answer: b. Write the ratio of the lengths of the second pair of corresponding legs. Answer: First find the length of AC using pythagorean theorem AC² + AB² = BC² AC² + 36 = 100 AC² = 64 AC = 8 Find the length of DF using pythagorean theorem DF² + DE² = EF² DF² + 18² = 30² DF² = 900 – 324 DF² = 576 DF= 24 c. Are these triangles similar? Does this suggest a Hypotenuse-Leg Similarity Theorem for right triangles? Explain. Answer: k = $$\frac { AC }{ DF }$$ = $$\frac { 8 }{ 24 }$$ = $$\frac { 1 }{ 3 }$$ k = $$\frac { AB }{ DE }$$ = $$\frac { 6 }{ 18 }$$ = $$\frac { 1 }{ 3 }$$ So, triangles are similar. Question 35. WRITING Can two triangles have all three ratios of corresponding angle measures equal to a value greater than 1 ? less than 1 ? Explain. Answer: Question 36. HOW DO YOU SEE IT? Which theorem could you use to show that ∆OPQ ~ ∆OMN in the portion of the Ferris wheel shown when PM = QN = 5 feet and MO = NO = 10 feet? Answer: The corresponding angle theorem states that ∆OPQ is similar to ∆OMN. Question 37. DRAWING CONCLUSIONS Explain why it is not necessary to have an Angle-Side-Angle Similarity Theorem. Answer: Question 38. THOUGHT PROVOKING Decide whether each is a valid method of showing that two quadrilaterals are similar. Justify your answer. a. SASA Answer: If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. b. SASAS Answer: If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent. c. SSSS Answer: If the corresponding side lengths of two triangles are proportional, then the triangles are similar. d. SASSS Answer: If two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then those two triangles are similar. Question 39. MULTIPLE REPRESENTATIONS Use a diagram to show why there is no Side-Side-Angle Similarity Theorem. Answer: Question 40. MODELING WITH MATHEMATICS The dimensions of an actual swing set are shown. You want to create a scale model of the swing set for a dollhouse using similar triangles. Sketch a drawing of your swing set and label each side length. Write a similarity statement for each pair of similar triangles. State the scale factor you used to create the scale model. Answer: Here we have to check the similarity statement for △ABC, △DEF. The scale factor k = $$\frac { AB }{ DE }$$ = $$\frac { 8 }{ 6 }$$ = $$\frac { 4 }{ 3 }$$ Question 41. PROVING A THEOREM Copy and complete the paragraph proot of the second part of the Slopes of Parallel Lines Theorem (Theorern 3. 13) from page 439. Given ml = mn, l and n are nonvertical. Prove l || n You are given that ml = mn. By the definition of slope. ml = $$\frac{B C}{A C}$$ and mn = $$\frac{E F}{D F}$$ By ____________, $$\frac{B C}{A C}=\frac{E F}{D F}$$. Rewriting this proportion yields ___________, By the Right Angles Congruence Theorem (Thin. 2.3), ___________, So. ∆ABC ~ ∆DEF by ___________ . Because corresponding angles of similar triangles are congruent, ∠BAC ≅∠EDF. By ___________, l || n. Answer: Question 42. PROVING A THEOREM Copy and complete the two-column proof 0f the second part of the Slopes of Perpendicular Lines Theorem (Theorem 3.14) Given ml mn = – 1, l and n are nonvertical. Prove l ⊥ n  Statements Reasons 1. mlmn = – 1 1. Given 2. ml = $$\frac{D E}{A D}$$, mn = $$\frac{A B}{B C}$$ 2. Definition of slope 3. $$\frac{D E}{A D} \cdot-\frac{A B}{B C}$$ = – 1 3. ________________________________ 4. $$\frac{D E}{A D}=\frac{B C}{A B}$$ 4. Multiply each side of statement 3 by –$$\frac{B C}{A B}$$. 5. $$\frac{D E}{B C}$$ = ____________ 5. Rewrite proportion. 6. ________________________________ 6. Right Angles Congruence Theorem (Thm. 2.3) 7. ∆ABC ~ ∆ADE 7. ________________________________ 8. ∠BAC ≅ ∠DAE 8. Corresponding angles of similar figures are congruent. 9. ∠BCA ≅ ∠CAD 9. Alternate Interior Angles Theorem (Thm. 3.2) 10. m∠BAC = m∠DAE, m∠BCA = m∠CAD 10. ________________________________ 11. m∠BAC + m∠BCA + 90° = 180° 11. ________________________________ 12. ________________________________ 12. Subtraction Property of Equality 13. m∠CAD + m∠DAE = 90° 13. Substitution Property of Equality 14. m∠CAE = m∠DAE + m∠CAD 14. Angle Addition Postulate (Post. 1.4) 15. m∠CAE = 90° 15. ________________________________ 16. ________________________________ 16. Definition of perpendicular lines Answer:  Statements Reasons 1. mlmn = – 1 1. Given 2. ml = $$\frac{D E}{A D}$$, mn = $$\frac{A B}{B C}$$ 2. Definition of slope 3. $$\frac{D E}{A D} \cdot-\frac{A B}{B C}$$ = – 1 3. Correspomsding sides are opposite 4. $$\frac{D E}{A D}=\frac{B C}{A B}$$ 4. Multiply each side of statement 3 by –$$\frac{B C}{A B}$$. 5. $$\frac{D E}{B C}$$ = $$\frac { AB }{ AD }$$ 5. Rewrite proportion. 6. Two right-angled triangles are said to be congruent to each other if the hypotenuse and one side of the right triangle are equal to the hypotenuse and the corresponding side of the other right-angled triangle. 6. Right Angles Congruence Theorem (Thm. 2.3) 7. ∆ABC ~ ∆ADE 7. According to the side angle side theorem. 8. ∠BAC ≅ ∠DAE 8. Corresponding angles of similar figures are congruent. 9. ∠BCA ≅ ∠CAD 9. Alternate Interior Angles Theorem (Thm. 3.2) 10. m∠BAC = m∠DAE, m∠BCA = m∠CAD 10. Congruent angles 11. m∠BAC + m∠BCA + 90° = 180° 11. △ABC is a right-angled triangle 12. m∠CAD + m∠DAE = 90° 12. Subtraction Property of Equality 13. m∠CAD + m∠DAE = 90° 13. Substitution Property of Equality 14. m∠CAE = m∠DAE + m∠CAD 14. Angle Addition Postulate (Post. 1.4) 15. m∠CAE = 90° 15. Right Angle 16. If two lines meet each other a an angle of 90°, then they are called the perpendicular lines. 16. Definition of perpendicular lines Maintaining Mathematical proficiency Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. Question 43. A(- 3, 6), B(2, 1); 3 to 2 Answer: Question 44. A(- 3, – 5), B(9, – 1); 1 to 3 Answer: Question 45. A(1, – 2), B(8, 12); 4 to 3 Answer: 8.4 Proportionality Theorems Exploration 1 Discovering a Proportionality Relationship Work with a partner. Use dynamic geometry software to draw any ∆ABC. a. Construct $$\overline{D E}$$ parallel to $$\overline{B C}$$ with endpoints on $$\overline{A B}$$ and $$\overline{A C}$$, respectively. Answer: b. Compare the ratios of AD to BD and AE to CE. Answer: c. Move $$\overline{D E}$$ to other locations Parallel to $$\overline{B C}$$ with endpoints on $$\overline{A B}$$ and $$\overline{A C}$$, and repeat part (b). Answer: d. Change ∆ABC and repeat parts (a) – (c) several times. Write a conjecture that summarizes your results. LOOKING FOR STRUCTURE To be proficient in math, you need to look closely to discern a pattern or structure. Answer: Exploration 2 Discovering a Proportionality Relationship Work with a partner. Use dynamic geometry software to draw any AABC. a. Bisect ∆B and plot point D at the intersection of the angle bisector and $$\overline{A C}$$. Answer: b. Compare the ratios of AD to DC and BA to BC. Answer: c. Change ∆ABC and repeat parts (a) and (b) several times. Write a conjecture that summarizes your results. Answer: Communicate Your Answer Question 3. What proportionality relationships exist in a triangle intersected by an angle bisector or by a line parallel to one of the sides? Answer: Question 4. Use the figure at the right to write a proportion. Answer: Lesson 8.4 Proportionality Theorems Monitoring Progress Question 1. Find the length of $$\overline{Y Z}$$. Answer: YZ = $$\frac { 315 }{ 11 }$$ Explanation: Triangle property thorem is $$\frac { XW }{ WV }$$ = $$\frac { XY }{ YZ }$$ $$\frac { 44 }{ 35 }$$ = $$\frac { 36 }{ YZ }$$ cross multiply the fractions 44 • YZ = 36 • 35 44 • YZ = 1260 YZ = $$\frac { 1260 }{ 44 }$$ YZ = $$\frac { 315 }{ 11 }$$ Question 2. Determine whether $$\overline{P S}$$ || $$\overline{Q R}$$ Answer: $$\frac { PQ }{ PN }$$ = $$\frac { 50 }{ 90 }$$ = $$\frac { 5 }{ 9 }$$ $$\frac { SR }{ SN }$$ = $$\frac { 40 }{ 72 }$$ = $$\frac { 5 }{ 9 }$$ $$\frac { PQ }{ PN }$$ = $$\frac { SR }{ SN }$$ so PS is parallel to QR Find the length of the given line segment. Question 3. $$\overline{B D}$$ Answer: $$\overline{B D}$$ = 12 Explanation: All the angles are congruent. So, $$\overline{A B}$$, $$\overline{C D}$$, $$\overline{E F}$$ are parallel. using the three parallel lines theorem $$\frac { BD }{ DF }$$ = $$\frac { AC }{ CE }$$ $$\frac { [latex]\overline{B D}$$ }{ 30 } = $$\frac { 16 }{ 40 }$$ $$\overline{B D}$$ = $$\frac { 16 }{ 40 }$$ • 30 $$\overline{B D}$$ = 12 Question 4. $$\overline{J M}$$ Answer: $$\overline{J M}$$ = $$\frac { 96 }{ 5 }$$ Explanation: All the angles are congruent. So, $$\overline{G H}$$, $$\overline{J K}$$, $$\overline{M N}$$ are parallel. using the three parallel lines theorem $$\frac { HK }{ KN }$$ = $$\frac { GJ }{ JM }$$ $$\frac { 15 }{ 18 }$$ = $$\frac { 16 }{ $\overline{J M}$$ }$ Cross multiply 15 • $$\overline{J M}$$ = 16 • 18 = 288 $$\overline{J M}$$ = $$\frac { 288 }{ 15 }$$ $$\overline{J M}$$ = $$\frac { 96 }{ 5 }$$ Find the value of the variable. Question 5. Answer: x = 28 Explanation: $$\overline{T V}$$ is the angle bisector So, $$\frac { ST }{ TU }$$ = $$\frac { SV }{ VU }$$ $$\frac { 14 }{ x }$$ = $$\frac { 24 }{ 48 }$$ cross multiply 24x = 14 • 48 = 672 x = $$\frac { 672 }{ 24 }$$ x = 28 Question 6. Answer: x = 4√2 Explanation: $$\overline{W Z}$$ is the angle bisector So, $$\frac { YZ }{ ZX }$$ = $$\frac { YW }{ WX }$$ $$\frac { 4 }{ 4 }$$ = $$\frac { 4√2 }{ x }$$ cross multiply 4x = 4 • 4√2 = 16√2 x = 4√2 Exercise 8.4 Proportionality Theorems Vocabulary and Core Concept Check Question 1. COMPLETE THE STATEMENT If a line divides two sides of a triangle proportionally, then it is ____________ to the third side. This theorem is knon as the ____________ . Answer: Question 2. VOCABULARY In ∆ABC, point R lies on $$\overline{B C}$$ and $$\vec{A}$$R bisects ∆CAB. Write the proportionality statement for the triangle that is based on the Triangle Angle Bisector Theorem (Theorem 8.9). Answer: According to the triangle angle bisector theorem $$\frac { CR }{ BR }$$ = $$\frac { AC }{ AB }$$ Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, find the length of $$\overline{A B}$$ . Question 3. Answer: Question 4. Answer: $$\frac { AE }{ ED }$$ = $$\frac { AB }{ BC }$$ $$\frac { 14 }{ 12 }$$ = $$\frac { AB }{ 18 }$$ AB = $$\frac { 14 }{ 12 }$$ • 18 AB = 21 units. In Exercises 5 – 8, determine whether $$\overline{K M}$$ || $$\overline{J N}$$. Question 5. Answer: Question 6. Answer: If $$\frac { JK }{ KL }$$ = $$\frac { NM }{ ML }$$, then KM || JN $$\frac { JK }{ KL }$$ = latex]\frac { 22.5 }{ 25 } [/latex] = latex]\frac { 9 }{ 10 } [/latex] $$\frac { NM }{ ML }$$ = $$\frac { 18 }{ 20 }$$ = latex]\frac { 9 }{ 10 } [/latex] $$\frac { JK }{ KL }$$ = $$\frac { NM }{ ML }$$ Hence KM || JN Question 7. Answer: Question 8. Answer: If $$\frac { JK }{ KL }$$ = $$\frac { NM }{ ML }$$, then KM || JN $$\frac { JK }{ KL }$$ = latex]\frac {35 }{ 16 } [/latex] $$\frac { NM }{ ML }$$ = $$\frac { 34 }{ 15 }$$ $$\frac { JK }{ KL }$$ ≠ $$\frac { NM }{ ML }$$ So, KM is not parallel to JN CONSTRUCTION In Exercises 9 – 12, draw a segment with the given length. Construct the point that divides the segment in the given ratio. Question 9. 3 in.; 1 to 4 Answer: Question 10. 2 in.; 2 to 3 Answer: Construct a 2 inch segment and divide the segment into 2 + 3 or 5 congruent pieces. Point P is the point that is $$\frac { 1 }{ 5 }$$ of the way from point A to point B. Question 11. 12 cm; 1 to 3 Answer: Question 12. 9 cm ; 2 to 5 Answer: Construct a 9 cm segment and divide the segment into 2 + 5 or 7 congruent pieces. Point p is the point that is $$\frac { 1 }{ 7 }$$ of the way from point A to point B. In Exercises 13 – 16, use the diagram to complete the proportion. Question 13. Answer: Question 14. Answer: $$\frac { CG }{ EG }$$ = $$\frac { BF }{ DF }$$ Question 15. Answer: Question 16. Answer: $$\frac { BF }{ BD }$$ = $$\frac { CG }{ CE }$$ In Exercises 17 and 18, find the length of the indicated line segment. Question 17. $$\overline{V X}$$ Answer: Question 18. $$\overline{S U}$$ Answer: $$\frac { SU }{ NS }$$ = $$\frac { RT }{ PR }$$ $$\frac { SU }{ 10 }$$ = $$\frac { 12 }{ 8 }$$ SU = $$\frac { 12 }{ 8 }$$ • 10 SU = 10 In Exercises 19 – 22, find the value of the variable. Question 19. Answer: Question 20. Answer: $$\frac { z }{ 1.5 }$$ = $$\frac { 3 }{ 4.5 }$$ z = $$\frac { 3 }{ 4.5 }$$ • 1.5 z = 1 Question 21. Answer: Question 22. Answer: $$\frac { q }{ 16 – q }$$ = $$\frac { 36 }{ 28 }$$ 28q = 36 (16 – q) 28q = 576 – 36q 28q + 36q = 576 64q = 576 q = 9 Question 23. ERROR ANALYSIS Describe and correct the error in solving for x. Answer: Question 24. ERROR ANALYSIS Describe and correct the error in the students reasoning. Answer: $$\frac { BD }{ CD }$$ = $$\frac { AB }{ AC }$$ BD = CD So, 1 = $$\frac { AB }{ AC }$$ AC = AB MATHEMATICAL CONNECTIONS In Exercises 25 and 26, find the value of x for which $$\overline{P Q}$$ || $$\overline{R S}$$. Question 25. Answer: Question 26. Answer: $$\frac { PR }{ RT }$$ = $$\frac { QS }{ ST }$$ $$\frac { 12 }{ 2x – 2 }$$ = $$\frac { 21 }{ 3x – 1 }$$ 12(3x – 1) = 21(2x – 2) 36x – 12 = 42x – 42 42x – 36x = 42 – 12 6x = 30 x = 5 Question 27. PROVING A THEOREM Prove the Triangle Proportionality Theorem (Theorem 8.6). Given $$\overline{Q S}$$ || $$\overline{T U}$$ Prove $$\frac{Q T}{T R}=\frac{S U}{U R}$$ Answer: Question 28. PROVING A THEOREM Prove the Converse of the Triangle Proportionality Theorem (Theorem 8.7). Given $$\frac{Z Y}{Y W}=\frac{Z X}{X V}$$ Prove $$\overline{Y X}$$ || $$\overline{W V}$$ Answer: Question 29. MODELING WITH MATHEMATICS The real estate term lake frontage refers to the distance along the edge of a piece of property that touches a lake. a. Find the lake frontage (to the nearest tenth) of each lot shown. b. In general, the more lake frontage a lot has, the higher its selling price. Which lot(s) should be listed for the highest price? c. Suppose that low prices are in the same ratio as lake frontages. If the least expensive lot is250,000, what are the prices of the other lots? Explain your reasoning.

Question 30.
USING STRUCTURE
Use the diagram to find the values of x and y.

$$\frac { 5 }{ 2 }$$ = $$\frac { x }{ 1.5 }$$
x = $$\frac { 5 }{ 2 }$$ • 1.5
x = 3.75
$$\frac { 3 }{ 7 }$$ = $$\frac { y }{ 5.25 }$$
y = $$\frac { 3 }{ 7 }$$ • 5.25
y = 2.25

Question 31.
REASONING
In the construction on page 447, explain why you can apply the Triangle Proportionality Theorem (Theorem 86) in Step 3.

Question 32.
PROVING A THEOREM
Use the diagram with the auxiliary line drawn to write a paragraph proof of the Three Parallel Lines Theorem (Theorem 8.8).
Given K1 || K2 || K3
Prove $$\frac{C B}{B A}=\frac{D E}{E F}$$

From the diagram, we can see that K₁ || K₂ || K₃
Those three parallel lines interest two traversals t₁, t₂
So, $$\frac{C B}{B A}=\frac{D E}{E F}$$

Question 33.
CRITICAL THINKING
In ∆LMN, the angle bisector of ∠M also bisects $$\overline{L N}$$. Classify ∆LMN as specifically as possible. Justify your answer.

Question 34.
HOW DO YOU SEE IT?
During a football game, the quarterback throws the ball to the receiver. The receiver is between two defensive players, as shown. If Player 1 is closer to the quarterback when the ball is thrown and both defensive players move at the same speed, which player will reach the receiver first? Explain your reasoning.

As per the image, player 1 is closer to the receiver. So, player 1 will reach the receiver first.

Question 35.
PROVING A THEOREM
Use the diagram with the auxiliary lines drawn to write a paragraph proof of the Triangle Angle Bisector Theorem (Theorem 8.9).
Given ∠YXW ≅ ∠WXZ
prove $$\frac{Y W}{W Z}=\frac{X Y}{X Z}$$

Question 36.
THOUGHT PROVOKING
Write the converse of the Triangle Angle Bisector Theorem (Theorem 8.9). Is the converse true? Justify your answer.

Question 37.
REASONING
How is the Triangle Midsegment Theorem (Theorem 6.8) related to the Triangle Proportionality Theorem (Theorem 8.6)? Explain your reasoning.

Question 38.
MAKING AN ARGUMENT
Two people leave points A and B at the same time. They intend to meet at point C at the same time. The person who leaves point A walks at a speed of 3 miles per hour. You and a friend are trying to determine how fast the person who leaves point B must walk. Your friend claims you need to know the length of $$\overline{A C}$$. Is your friend correct? Explain your reasoning.

My self starts walking from point A with a speed of 3 miles per hour and reaches point C.
My friend starts walking from point B with x speed and reaches point C.
$$\frac { AD }{ DC }$$ = $$\frac { BE }{ CE }$$
I have to travel from A to C. So, I need to know distance between AC.
Therefore, my friend is correct.

Question 39.
CONSTRUCTION
Given segments with lengths r, s, and t, construct a segment of length x, such that $$\frac{r}{s}=\frac{t}{x}$$

Question 40.
PROOF
Prove Ceva’s Theorem: If P is any point inside ∆ABC, then $$\frac{A Y}{Y C} \cdot \frac{C X}{X B} \cdot \frac{B Z}{Z A}$$ = 1

(Hint: Draw segments parallel to $$\overline{B Y}$$ through A and C, as shown. Apply the Triangle Proportionality Theorem (Theorem 8.6) to ∆ACM. Show that ∆APN ~ ∆MPC, ∆CXM ~ ∆BXP, and ∆BZP ~ ∆AZN.)

Maintaining Mathematical Proficiency

Use the triangle.

Question 41.
Which sides are the legs?

Question 42.
Which side is the hypotenuse?
The leg c is the hypotenuse.

Solve the equation.

Question 43.
x2 = 121

Question 44.
x2 + 16 = 25
x² + 16 = 25
x² = 25 – 16
x² = 9
x = 3

Question 45.
36 + x2 = 85

8.1 Similar Polygons

Find the scale factor. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality.

Question 1.
ABCD ~ EFGH

$$\frac { BC }{ CD }$$ = $$\frac { 8 }{ 12 }$$ = $$\frac { 2 }{ 3 }$$
$$\frac { EH }{ GH }$$ = $$\frac { 6 }{ 9 }$$ = $$\frac { 2 }{ 3 }$$
So, scale factor = $$\frac { 2 }{ 3 }$$

Question 2.
∆XYZ ~ ∆RPQ

longer sides: $$\frac { 10 }{ 25 }$$ = $$\frac { 2 }{ 5 }$$
shorter sides: $$\frac { 6 }{ 15 }$$ = $$\frac { 2 }{ 5 }$$
remaining sides: $$\frac { 8 }{ 20 }$$ = $$\frac { 2 }{ 5 }$$
So, scale factor = $$\frac { 2 }{ 5 }$$

Question 3.
Two similar triangles have a scale factor of 3 : 5. The altitude of the larger triangle is 24 inches. What is the altitude of the smaller triangle?
Scale factor of smaller triangle to larger traingle is $$\frac { 3 }{ 5 }$$ and larger traingle altitude is 24 inches
Let x be the smaller triangle altitude
$$\frac { altitude of smaller triangle }{ altitude of larger traingle }$$ = scale factor
$$\frac { x }{ 24 }$$ = $$\frac { 3 }{ 5 }$$
x = $$\frac { 3 }{ 5 }$$ • 24
x = 14.4

Question 4.
Two similar triangles have a pair of corresponding sides of length 12 meters and 8 meters. The larger triangle has a perimeter of 48 meters and an area of 180 square meters. Find the perimeter and area of the smaller triangle.
Scale factor = $$\frac { 2 }{ 3 }$$
perimeter of smaller triangle = 32
Area of smaller triangle = 80

Explanation:
The scale factor of smaller to larger traingle = $$\frac { 8 }{ 12 }$$ = $$\frac { 2 }{ 3 }$$
$$\frac { perimeter of smaller triangle }{ perimeter of larger triangle }$$ = scale factor$$\frac { perimeter of smaller triangle }{ 48 }$$ = $$\frac { 2 }{ 3 }$$
perimeter of smaller triangle = $$\frac { 2 }{ 3 }$$ • 48
perimeter of smaller triangle = 32
$$\frac { Area of smaller triangle }{ Area of larger triangle }$$ = (scale factor)²
$$\frac { Area of smaller triangle }{ 180 }$$ = ( $$\frac { 2 }{ 3 }$$)²
Area of smaller triangle = $$\frac { 4 }{ 9 }$$ • 180
= 80

8.2 Proving Triangle Similarity by AA

Show that the triangles are similar. Write a similarity statement.

Question 5.

m∠RQS = m∠UTS = 30°.
△QRS and △STU are similar as per the AA similarity theorem.

Question 6.

m∠CAB = 60°, m∠DEF = 30°
△ABC and △DEF are not similar as per the AA similarity theorem.

Question 7.
A cellular telephone tower casts a shadow that is 72 feet long, while a nearby tree that is 27 feet tall casts a shadow that is 6 feet long. How tall is the tower?
$$\frac { shadow of tree }{ shadow of tower }$$ = $$\frac { height of tree }{ height of tower }$$
$$\frac { 6 }{ 72 }$$ = $$\frac { 27 }{ x }$$
6x = 1944
x = 324 ft
The height of the tower is 324 ft.

8.3 Proving Triangle Similarity by SSS and SAS

Use the SSS Similarity Theorem (Theorem 8.4) or the SAS Similarity Theorem (Theorem 8.5) to show that the triangles are similar.

Question 8.

$$\frac { DE }{ CD }$$ = $$\frac { 7 }{ 3.5 }$$ = 2
$$\frac { AB }{ BC }$$ = $$\frac { 8 }{ 4 }$$ = 2
$$\frac { DE }{ CD }$$ = $$\frac { AB }{ BC }$$
So, BD is parallel to AE.

Question 9.

$$\frac { QU }{ TU }$$ = $$\frac { 9 }{ 4.5 }$$ = 2
$$\frac { QR }{ SR }$$ = $$\frac { 14 }{ 7 }$$ = 2
$$\frac { QU }{ TU }$$ = $$\frac { QR }{ SR }$$
So, ST is parallel to RU.

Question 10.
Find the value of x that makes ∆ABC ~ ∆DEF

$$\frac { 24 }{ 6 }$$ = 4
$$\frac { 32 }{ 2x }$$ = 4
32 = 8x
x = $$\frac { 32 }{ 8 }$$
x = 4

8.4 Proportionality Theorems

Determine whether $$\overline{A B}$$ || $$\overline{C D}$$

Question 11.

$$\frac { DB }{ BE }$$ = $$\frac { 10 }{ 16 }$$ = $$\frac { 5 }{ 8 }$$
$$\frac { CA }{ AE }$$ = $$\frac { 20 }{ 28 }$$ = $$\frac { 5 }{ 7 }$$
$$\frac { DB }{ BE }$$ ≠ $$\frac { CA }{ AE }$$
So, CD and AB are not parallel.

Question 12.

$$\frac { DB }{ BE }$$ = $$\frac { 12 }{ 20 }$$ = $$\frac { 3 }{ 5 }$$
$$\frac { CA }{ AE }$$ = $$\frac { 13.5 }{ 22.5 }$$ = $$\frac { 3 }{ 5 }$$
$$\frac { DB }{ BE }$$ = $$\frac { CA }{ AE }$$
So, AB and CD are parallel.

Question 13.
Find the length of $$\overline{Y B}$$.

$$\frac { ZC }{ AZ }$$ = $$\frac { 24 }{ 15 }$$ = $$\frac { 8 }{ 5 }$$
$$\frac { YB }{ AY }$$ = $$\frac { ZC }{ AZ }$$
$$\frac { YB }{ 7 }$$ = $$\frac { 8 }{ 5 }$$
YB = $$\frac { 8 }{ 5 }$$ • 7
YB = $$\frac { 56 }{ 5 }$$
The length of $$\overline{Y B}$$ is $$\frac { 56 }{ 5 }$$

Find the length of $$\overline{A B}$$.

Question 14.

$$\frac { AB }{ 7 }$$ = $$\frac { 6 }{ 4 }$$
$$\overline{A B}$$ = $$\frac { 6 }{ 4 }$$ • 7
$$\overline{A B}$$ = $$\frac { 42 }{ 4 }$$
$$\overline{A B}$$ = $$\frac { 21 }{ 2 }$$
The length of $$\overline{A B}$$ is $$\frac { 21 }{ 2 }$$.

Question 15.

$$\frac { DB }{ CD }$$ = $$\frac { AB }{ AC }$$
$$\frac { 4 }{ 10 }$$ = $$\frac { AB }{ 18 }$$
$$\frac { 2 }{ 5 }$$ = $$\frac { AB }{ 18 }$$
AB = $$\frac { 36 }{ 5 }$$

Similarity Test

Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning.

Question 1.

Longer sides: $$\frac { 32 }{ 24 }$$ = $$\frac { 4 }{ 3 }$$
shorter sides: $$\frac { 18 }{ 14 }$$ = $$\frac { 9 }{ 7 }$$
remaining sides: $$\frac { 20 }{ 15 }$$ = $$\frac { 4 }{ 3 }$$
Those are not equal.
So, triangles are not similar.

Question 2.

$$\frac { AC }{ KJ }$$ = $$\frac { 6 }{ 8 }$$ = $$\frac { 3 }{ 4 }$$
$$\frac { BC }{ JL }$$ = $$\frac { 8 }{ $\frac { 32 }{ 3 }$$ }$ = $$\frac { 3 }{ 4 }$$
$$\frac { AC }{ KJ }$$ = $$\frac { BC }{ JL }$$
∠C = ∠J
So, △ABC and △JLK are similar.

Question 3.

$$\frac { XY }{ XW }$$ = $$\frac { PZ }{ PW }$$

Find the value of the variable.

Question 4.

$$\frac { 9 }{ w }$$ = $$\frac { 15 }{ 5 }$$
$$\frac { 9 }{ w }$$ = 3
9 = 3w
w = 3

Question 5.

$$\frac { 17.5 }{ 21 }$$ = $$\frac { q }{ 33 }$$
q = $$\frac { 17.5 }{ 21 }$$ • 33
q = $$\frac { 55 }{ 2 }$$

Question 6.

$$\frac { 21 – p }{ p }$$ = $$\frac { 12 }{ 24 }$$
$$\frac { 21 – p }{ p }$$ = $$\frac { 1 }{ 2 }$$
cross multiply
2(21 – p) = p
42 – 2p = p
42 = p + 2p
42 = 3p
p = 14

Question 7.
Given ∆QRS ~ ∆MNP, list all pairs of congruent angles, Then write the ratios of the corresponding side lengths in a statement of proportionality.

The pairs of congruent anglres are m∠QRS, m∠RSQ, m∠SQR, m∠MNP, m∠NPM, m∠PMN.
The ratios of side lengths are $$\frac { RQ }{ MN }$$, $$\frac { QS }{ MP }$$, $$\frac { RS }{ NP }$$

Use the diagram.

Question 8.
Find the length of $$\overline{E F}$$.

$$\frac { DE }{ EF }$$ = $$\frac { CD }{ BC }$$
$$\frac { 3.2 }{ EF }$$ = $$\frac { 2.8 }{ 1.4 }$$
EF = 1.6
The length of $$\overline{E F}$$ is 1.6

Question 9.
Find the length of $$\overline{F G}$$.

$$\frac { EF }{ FG }$$ = $$\frac { BC }{ AB }$$
$$\frac { 1.6 }{ FG }$$ = $$\frac { 1.4 }{ 4.2 }$$
FG = 4.8
The length of $$\overline{F G}$$ is 4.8

Question 10.
Is quadrilateral FECB similar to quadrilateral GFBA? If so, what is the scale factor of the dilation that maps quadrilateral FECB to quadrilateral GFBA?

The scale factor of dilation from quadrilateral FECB to quadrilateral GFBA is $$\frac { GF }{ FE }$$

Question 11.
You are visiting the Unisphere at Flushing Meadows Corona Park in New York. To estimate the height of the stainless steel model of Earth. you place a mirror on the ground and stand where you can see the top of the model in the mirror. Use the diagram to estimate the height of the model. Explain why this method works.

Question 12.
You are making a scale model of a rectangular park for a school project. Your model has a length of 2 feet and a width of 1.4 feet. The actual park is 800 yards long. What are the perimeter and area of the actual park?

we know that 1 yard = ft
As per the similarity theorem
$$\frac { AD }{ EH }$$ = $$\frac { AB }{ EF }$$
$$\frac { 2 }{ 800.3 }$$ = $$\frac { 1.4 }{ EF }$$
EF = 1200 • 1 • 4
EF = 1680
Perimete of the park P = 2(1680 + 2400)
= 2(4080) = 8160 ft
Area of the actual park = 1680 • 2400 = 4,032,000 sq ft
Therefore, perimeter of the actual park = 8160 ft
area of the actual park is 4,032,000 sq ft.

Question 13.
In a Perspective drawing, lines that are parallel in real life must meet at a vanishing point on the horizon. To make the train cars in the drawing appear equal in length, they are drawn so that the lines connecting the opposite corners of each car are parallel. Use the dimensions given and the yellow parallel lines to find the length of the bottom edge of the drawing of Car 2.

$$\frac { 5.4 }{ 10.6 }$$ = $$\frac { 5.4 + C }{ 19 }$$
C = $$\frac { 45.36 }{ 10.6 }$$
$$\frac { (19 – x – 8.4) }{ (19 – 8.4) }$$ = $$\frac { 5.4 }{ 5.4 + c }$$
$$\frac { 19 – 8.4 }{ 19 }$$ = $$\frac { c2 + 5.4 }{ c1 + c2 + 5.4 }$$
$$\frac { 19 – 8.4 }{ 19 }$$ = $$\frac { 5.4 }{ 5.4 + c2 }$$
c1 = 7.6, c2 = 4.2, x = 4.6
The length of car 2 is 4.2 cm.

Similarity Cumulative Assessment

Question 1.
Use the graph of quadrilaterals ABCD and QRST.

a. Write a composition of transformations that maps quadrilateral ABCD to quadrilateral QRST.
The scale factor = $$\frac { AD }{ QT }$$ = $$\frac { 2 }{ 1.5 }$$

No.
$$\frac { AD }{ QT }$$ = $$\frac { 2 }{ 1.5 }$$
$$\frac { CD }{ TS }$$ = $$\frac { 2.8 }{ 1.4 }$$ = 2

Question 2.
In the diagram. ABCD is a parallelogram. Which congruence theorem(s) could you Use to show that ∆AED ≅ ∆CEB? Select all that apply.

SAS Congruence Theorem (Theorem 5.5)
It states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent.

SSS Congruence Theorem (Theorem 5.8)
If all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent by SSS rule.

HL Congruence Theorem (Theorem 5.9)
A given set of triangles are congruent if the corresponding lengths of their hypotenuse and one leg are equal.

ASA Congruence Theorem (Theorem 5. 10)
If any two angles and the side included between the angles of one triangle are equivalent to the corresponding two angles and side included between the angles of the second triangle, then the two triangles are said to be congruent by ASA rule

AAS Congruence Theorem (Theorem 5. 11)
AAS stands for Angle-angle-side. When two angles and a non-included side of a triangle are equal to the corresponding angles and sides of another triangle, then the triangles are said to be congruent.

Question 3.
By the Triangle Proportionality Theorem (Theorem 8.6), $$\frac{V W}{W Y}=\frac{V X}{X Z}$$ In the diagram, VX > VW and XZ > WY. List three possible values for VX and XZ.

$$\frac{V W}{W Y}=\frac{V X}{X Z}$$
$$\frac{ 4 }{6}=\frac{V X}{X Z}$$
The possible values of VX are greater than 4 means 5, 6, 7, . . .
The possible values of XZ are greater than 6 means 7, 8, 9, . .

Question 4.
The slope of line l is – $$\frac{3}{4}$$. The slope of line n is $$\frac{4}{3}$$ What must be true about lines l and n ?
(A) Lines l and n are parallel.
(B) Lines l and n arc perpendicular.
(C) Lines l and n are skew.
(D) Lines l and n are the same line.
The slope of l = – $$\frac{3}{4}$$
Slope of n = $$\frac{4}{3}$$
lines slopes are reciprocal and opposite. So, they are perpendicular.

Question 5.
Enter a statement or reason in each blank to complete the two-column proof.

Given $$\frac{K J}{K L}=\frac{K H}{K M}$$
Prove ∠LMN ≅ ∠JHG

 Statements Reasons 1. $$\frac{K J}{K L}=\frac{K H}{K M}$$ 1. Given 2. ∠JKH ≅ ∠LKM 2. ________________________ 3. ∆JKH ~ ∆LKM 3. ________________________ 4. ∠KHJ ≅∠KML 4. ________________________ 5. _______________________ 5. Definition of congruent angles 6. m∠KHJ + m∠JHG = 180° 6. Linear Pair Postulate (Post. 18) 7. m∠JHG = 180° – m∠KHJ 7. ________________________ 8. m∠KML + m∠LMN = 180° 8. ________________________ 9. ________________________ 9. Subtraction Property of Equality 10. m∠LMN = 180° – m∠KHJ 10. ________________________ 11. ________________________ 11. Transitive Property of Equality 12. ∠LMN ≅ ∠JHG 12. ________________________

Question 6.
The coordinates of the vertices of ∆DEF are D(- 8, 5), E(- 5, 8), and F(- 1, 4), The coordinates of the vertices of ∆JKL are J(16, – 10), K(10, – 16), and L(2, – 8), ∠D ≅ ∠J. Can you show that ∆DEF ∆JKL by using the AA Similarity Theorem (Theorem 8.3)? If so, do so by listing the congruent corresponding angles and writing a similarity transformation that maps ∆DEF to ∆JKL. If not, explain why not.
AA similarity theorem states that ∠D = ∠J. So, ∆DEF and ∆JKL are similar.

Question 7.
Classify the quadrilateral using the most specific name.

rectangle     square    parallelogram    rhombus

Question 8.
‘Your friend makes the statement “Quadrilateral PQRS is similar to quadrilateral WXYZ.” Describe the relationships between corresponding angles and between corresponding sides that make this statement true.

When 2 figures are similar, then their corresponding angles are congruent and their corresponding lengths are proportional. hence if PQRS is similar to wxyZ, then the following statements are true.
∠P = ∠W, ∠Q = ∠X, ∠R = ∠Y and ∠S = ∠Z and
$$\frac { PQ }{ WX }$$ = $$\frac { QR }{ XY }$$ = $$\frac { RS }{ YZ }$$ = $$\frac { PS }{ WZ }$$ = k
Here is a constant of proportionality.

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Big Ideas Math Book Algebra 2 Answer Key Chapter 2 Quadratic Functions

Solving all homework & assignments questions can be simple by referring to this BIM Book Algebra 2 Answer Key. Also, you can learn all the complex concepts of Chapter 2 Quadratic Functions with the help of the Big Ideas Math Algebra 2 Book Solution Key and can answer any type of questions related to Chapter 2 Quadratic Functions complex topics. Make use of these below available BIM Book Algebra 2 Answer Key Chapter 2 Quadratic Functions direct links and prepare well to score good results in any examinations.

Find the x-intercept of the graph of the linear equation.

Question 1.
y = 2x + 7

Question 1.
y = 2x + 7

Question 2.
y = -6x + 8

Question 3.
y = -10x – 36

Question 4.
y = 3(x – 5)

Question 5.
y = -4(x + 10)

Question 6.
3x + 6y = 24

Find the distance between the two points.

Question 7.
(2, 5), (-4, 7)

Question 8.
(-1, 0), (-8, 4)

Question 9.
(3, 10), (5, 9)

Question 10.
(7, -4), (-5, 0)

Question 11.
(4, -8), (4, 2)

Question 12.
(0, 9), (-3, -6)

Question 13.
ABSTRACT REASONING Use the Distance Formula to write an expression for the distance between the two points (a, c) and (b, c). Is there an easier way to find the distance when the x-coordinates are equal? Explain your reasoning

Monitoring Progress

Decide whether the syllogism represents correct or flawed reasoning. If flawed, explain why the conclusion is not valid.

Question 1.
All mammals are warm-blooded.
All dogs are mammals.
Therefore, all dogs are warm-blooded.

Question 2.
All mammals are warm-blooded.
My pet is warm-blooded.
Therefore, my pet is a mammal.

Question 3.
If I am sick, then I will miss school.
I missed school.
Therefore, I am sick.

Question 4.
If I am sick, then I will miss school.
I did not miss school.
Therefore, I am not sick.

Lesson 2.1 Transformations of Quadratic Functions

Essential Question

How do the constants a, h, and k affect the graph of the quadratic function g(x) = a(x – h)2 + k?
The parent function of the quadratic family is f(x) = x2. A transformation of the graph of the parent function is represented by the function g(x) = a(x – h)2 + k, where a ≠ 0.

EXPLORATION 1
Work with a partner.
a. g(x) = -(x – 2)2
b. g(x) = (x – 2)2 + 2
c. g(x) = -(x + 2)2 – 2
d. g(x) = 0.5(x – 2)2 + 2
e. g(x) = 2(x – 2)2
f. g(x) = -(x + 2)2 + 2

Question 2.
How do the constants a, h, and k affect the graph of the quadratic function g(x) =a(x – h)2 + k?

Question 3.
Write the equation of the quadratic function whose graph is shown at the right. Explain your reasoning. Then use a graphing calculator to verify that your equation is correct.

2.1 Lesson

Monitoring Progress

Describe the transformation of f(x) = x2 represented by g. Then graph each function.

Question 1.
g(x) = (x – 3)2

Question 2.
g(x) = (x + 2)2 – 2

Question 3.
g(x) = (x + 5)2 + 1

Describe the transformation of f(x) = x2 represented by g. Then graph each function.

Question 4.
g(x) = ($$\frac{1}{3} x$$)2

Question 5.
g(x) = 3(x – 1)2

Question 6.
g(x) = -(x + 3)2 + 2

Question 7.
Let the graph of g be a vertical shrink by a factor of $$\frac{1}{2}$$ followed by a translation 2 units up of the graph of f(x) = x2. Write a rule for g and identify the vertex.

Question 8.
Let the graph of g be a translation 4 units left followed by a horizontal shrink by a factor of $$\frac{1}{3}$$ of the graph of f(x) = x2 + x. Write a rule for g.

Question 9.
WHAT IF? In Example 5, the water hits the ground 10 feet closer to the fire truck after lowering the ladder. Write a function that models the new path of the water.

Transformations of Quadratic Functions 2.1 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE The graph of a quadratic function is called a(n) ________.

Question 2.
VOCABULARY Identify the vertex of the parabola given by f(x) = (x + 2)2 – 4.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–12, describe the transformation of f(x) = x2 represented by g. Then graph each function.

Question 3.
g(x) = x2 – 3

Question 4.
g(x) = x2 + 1

Question 5.
g(x) = (x + 2)2

Question 6.
g(x) = (x – 4)2

Question 7.
g(x) = (x – 1)2

Question 8.
g(x) = (x + 3)2

Question 9.
g(x) = (x + 6)2 – 2

Question 10.
g(x) = (x – 9)2 + 5

Question 11.
g(x) = (x – 7)2 + 1

Question 12.
g(x) = (x + 10)2 – 3

ANALYZING RELATIONSHIPS In Exercises 13–16, match the function with the correct transformation of the graph of f. Explain your reasoning.

Question 13.
y = f(x – 1)

Question 14.
y = f(x) + 1

Question 15.
y = f(x – 1) + 1

Question 16.
y = f(x + 1)

In Exercises 17–24, describe the transformation of f(x) = x2 represented by g. Then graph each function.

Question 17.
g(x) = -x2

Question 18.
g(x) = (-x)2

Question 19.
g(x) = 3x2

Question 20.
g(x) = $$\frac{1}{3}$$x2

Question 21.
g(x) = (2x)2

Question 22.
g(x) = -(2x)2

Question 23.
g(x) = $$\frac{1}{5}$$x2 – 4

Question 24.
g(x) = $$\frac{1}{2}$$(x – 1)2

ERROR ANALYSIS In Exercises 25 and 26, describe and correct the error in analyzing the graph of f(x) = −6x2 + 4.

Question 25.

Question 26.

USING STRUCTURE In Exercises 27–30, describe the transformation of the graph of the parent quadratic function. Then identify the vertex.

Question 27.
f(x) = 3(x + 2)2 + 1

Question 28.
f(x) = -4(x + 1)2 – 5

Question 29.
f(x) = -2x2 + 5

Question 30.
f(x) = $$\frac{1}{2}$$(x – 1)2

In Exercises 31–34, write a rule for g described by the transformations of the graph of f. Then identify the vertex.

Question 31.
f(x) = x2 vertical stretch by a factor of 4 and a reflection in the x-axis, followed by a translation 2 units up

Question 32.
f(x) = x2; vertical shrink by a factor of $$\frac{1}{3}$$ and a reflection in the y-axis, followed by a translation 3 units right

Question 33.
f(x) = 8x2 – 6; horizontal stretch by a factor of 2 and a translation 2 units up, followed by a reflection in the y-axis

Question 34.
f(x) = (x + 6)2 + 3; horizontal shrink by a factor of $$\frac{1}{2}$$ and a translation 1 unit down, followed by a reflection in the x-axis

USING TOOLS In Exercises 35–40, match the function with its graph. Explain your reasoning.

Question 35.
g(x) = 2(x – 1)2 – 2

Question 36.
g(x) = $$\frac{1}{2}$$(x + 1)2 – 2

Question 37.
g(x) = -2(x – 1)2 + 2

Question 38.
g(x) = 2(x + 1)2 + 2

Question 39.
g(x) = -2(x + 1)2 – 2

Question 40.
g(x) = 2(x – 1)2 + 2

JUSTIFYING STEPS In Exercises 41 and 42, justify eachstep in writing a function g based on the transformationsof f(x) = 2x2 + 6x.

Question 41.
translation 6 units down followed by a reflection in the x-axis

Question 42.
reflection in the y-axis followed by a translation 4 units right

Question 43.
MODELING WITH MATHEMATICS The function h(x) = -0.03(x – 14)2 + 6 models the jump of a red kangaroo, where x is the horizontal distance traveled (in feet) and h(x) is the height (in feet). When the kangaroo jumps from a higher location, it lands 5 feet farther away. Write a function that models the second jump.

MODELING WITH MATHEMATICS The function f(t) = -16t2 + 10 models the height (in feet) of an object t seconds after it is dropped from a height of 10 feet on Earth. The same object dropped from the same height on the moon is modeled by g(t) = –$$\frac{8}{3}$$t2 + 10. Describe the transformation of the graph of f to obtain g. From what height must the object be dropped on the moon so it hits the ground at the same time as on Earth?