Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions

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

The solutions seen in Big Ideas Math Book Algebra 2 Answer Key is prepared by math professionals in a very simple manner with explanations. With the help of BIM Algebra 2 Answer Key students can score good grades in any of their exams and can make you achieve what you are looking for but improving their subject knowledge. Before that we suggest you to check the list of the chapters covered in BIM Algebra 2 Answer Key Ch 1 Linear Functions. Big Ideas Math Book Algebra 2 Chapter 1 Solutions is the best guide for you to score well in exams. Just tap the links provided below and start solving the questions.

• Linear Functions Maintaining Mathematical Proficiency – Page 1
• Linear Functions Maintaining Mathematical Practices – Page 2
• Lesson 1.1 Parent Functions and Transformations – Page (4-10)
• Parent Functions and Transformations 1.1 Exercises – Page (8-10)
• Lesson 1.2 Transformations of Linear and Absolute Value Functions – Page (12-18)
• Transformations of Linear and Absolute Value Functions 1.2 Exercises – Page (16-18)
• Linear Functions Study Skills Taking Control of Your Class Time – Page 19
• Linear Functions 1.1-1.2 Quiz – Page 20
• Lesson 1.3 Modeling with Linear Functions – Page (22-28)
• Modeling with Linear Functions 1.3 Exercises – Page (26-28)
• Lesson 1.4 Solving Linear Systems – Page (30-36)
• Solving Linear Systems 1.4 Exercises – Page (34-36)
• Linear Functions Performance Task: Secret of the Hanging Baskets – Page 37
• Linear Functions Chapter Review – Page (38-40)
• Linear Functions Chapter Test – Page 41
• Linear Functions Cumulative Assessment – Page (42-43)

Linear Functions Maintaining Mathematical Proficiency

Evaluate.

Question 1.
5 • 2³ + 7
Answer: 47

Explanation:
Given expression,
5 • 2³ + 7
= 5.8+7
= 40+7 = 47
So, the expression 5 • 2³ + 7 = 47.

Question 2.
4 – 2(3 + 2)²
Answer: -46

Explanation:
Given expression,
4 – 2(3 + 2)²
= 4 – 2 (25)
= 4  – 2 (25)
= 4 – 50
= – 46
So, the expression = 4 – 2(3 + 2)² = -46

Question 3.
48 ÷ 4² + \(\frac{3}{5}\)
Answer: 3.6

Explanation:
Given expression,
48 ÷ 4² + \(\frac{3}{5}\)
= 48 ÷ 16 + \(\frac{3}{5}\)
= 3 + \(\frac{3}{5}\) = 3.6
So, 48 ÷ 4² + \(\frac{3}{5}\) = 3.6

Question 4.
50 ÷ 5² • 2
Answer: 4

Explanation:
50 ÷ 5² • 2
= 50 ÷ 25 . 2
= 2 .2
= 4
So, 50 ÷ 5² • 2 = 4

Question 5.
\(\frac{1}{2}\)(2²+ 22)
Answer: 13

Explanation:
Given,
\(\frac{1}{2}\)(2²+ 22)
\(\frac{1}{2}\)(4+ 22)
\(\frac{1}{2}\)(26) = 13
So, \(\frac{1}{2}\)(2²+ 22) is 13.

Question 6.
\(\frac{1}{6}\)(6 + 18) – 2²
Answer: 0

Explanation:
Given,
\(\frac{1}{6}\)(6 + 18) – 2²
\(\frac{1}{6}\)(24) – 4
4 – 4 = 0
So, \(\frac{1}{6}\)(6 + 18) – 2² = 0

Graph the transformation of the figure.

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

Answer:

We have to draw a rectangle 1 unit right and 4 units up. So, start drawing the rectangle from the origin 0 to 3.

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

Answer:

Start drawing the triangle 2 units left.

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

Answer:

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.
Answer:
The order of operations says the order to solve steps in expressions with more than one operation. First, we solve any operations inside 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
Answer:
Given equation is y = 2x – 3
When x = 0
y = 2(0) – 3 = -3
y = -3
x = 1
y = 2(1) – 3 = 2 – 3 = -1
x = 2
y = 2(2) – 3 = 1
x = 3
y = 2(3) – 3 = 6 – 3 = 3

Question 2.
y = | x + 2 |
Answer:
Given equation is y = | x + 2 |
x = -2
y = |-2 + 2| = 0
x = -1
y = |-1 + 2| = 1
x = 0
y = |0 + 2| = 2
x = 1
y = |1 + 2| = 3
x = 2
y = |2 + 2| = 4

Question 3.
y = -x² + 1
Answer:
Given equation is y = -x² + 1
x = -1
y = -x² + 1
y = -1 + 1 = 0
x = 0
y = 0 + 1 = 1
x = 1
y = -1 + 1 = 0

Question 4.
y = \(\sqrt{x-1}\)
Answer:
Given equation is y = \(\sqrt{x-1}\)

Question 5.
y = x³ – 2
Answer:
Given equation is y = x³ – 2
x = 0
y = 0 – 2 = -2
(0, -2)
x = 1
y = 1 – 2 = -1
(1, -1)
x = -1
y = -1 – 2 = -3
(-1, -3)
x = 2
y = 2³ – 2
y = 8 – 2 = 6
(2, 6)

Question 6.
y = 0.25x³
Answer:
Given equation is y = 0.25x³
x = -1
y = 0.25(-1)³
y = -0.25
x = 0
y = 0.25(0)
y = 0
x = 1
y = 0.25(1)³
y = 0.25
x = 2
y = 0.25(2)³
y = 0.25(8)
y = 2
(2, 2)

Determine whether the viewing window is square. Explain.

Question 7.
-8 ≤ x ≤ 8, -2 ≤ y ≤ 8
Answer: Square

Explanation:
Given,
-8 ≤ x ≤ 8, -2 ≤ y ≤ 8
The total range of the X-axis is 16 units and the total range of the Y-axis is 10 units
The ratio of the height to width of the viewing screen is \(\frac{10}{16}\) = \(\frac{5}{8}\)
So, the ratio is 5:8.
Hence the viewing window is square.

Question 8.
-7 ≤ x ≤ 8, -2 ≤ y ≤ 8
Answer: Square

Explanation:
Given,
-7 ≤ x ≤ 8, -2 ≤ y ≤ 8
The total range of the X-axis is 15 units and the total range of the Y-axis is 10 units
The ratio of the height to width of the viewing screen is \(\frac{10}{15}\) = \(\frac{2}{3}\)
So, the ratio is 2:3.
Hence the viewing window is square.

Question 9.
-6 ≤ x ≤ 9, -2 ≤ y ≤ 8
Answer: Square

Explanation:
Given,
-6 ≤ x ≤ 9, -2 ≤ y ≤ 8
The total range of the X-axis is 15 units and the total range of the Y-axis is 10 units
The ratio of the height to width of the viewing screen is \(\frac{10}{15}\) = \(\frac{2}{3}\)
So, the ratio is 2:3.
Thus the viewing window is square.

Question 10.
-2 ≤ x≤ 2, -3 ≤ y ≤ 3
Answer: not a square

Explanation:
Given,
-2 ≤ x≤ 2, -3 ≤ y ≤ 3
The total range of the X-axis is 4 units and the total range of the Y-axis is 6 units
The ratio of the height to width of the viewing screen is \(\frac{6}{4}\) = \(\frac{3}{2}\)
So, the ratio is 3:2.
The viewing window is not a square.

Question 11.
-4 ≤ x ≤ 5, -3 ≤ y ≤ 3
Answer: square

Explanation:
Given,
-4 ≤ x ≤ 5, -3 ≤ y ≤ 3
The total range of the X-axis is 9 units and the total range of the Y-axis is 6 units
The ratio of the height to width of the viewing screen is \(\frac{6}{9}\) = \(\frac{2}{3}\)
So, the ratio is 2:3.
The viewing window is a square.

Question 12.
-4 ≤ x ≤ 4, -3 ≤ y ≤ 3
Answer: square

Explanation:
Given,
-4 ≤ x ≤ 4, -3 ≤ y ≤ 3
The total range of the X-axis is 8 units and the total range of the Y-axis is 6 units
The ratio of the height to width of the viewing screen is \(\frac{6}{8}\) = \(\frac{2}{3}\)
So, the ratio is 2:3.
The viewing window is a square.

Lesson 1.1 Parent Functions and Transformations

Essential Question

What are the characteristics of some of the basic parent functions?
Answer:
Odd. End behavior goes in different directions.
If a function is positive, the left side of the graph will point down and the right side will point up i.e., 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.

Communicate Your Answer

Question 2.
What are the characteristics of some of the basic parent functions?
Answer: The key common points of linear parent functions include the fact that the: Equation is y = x. Domain and range are real numbers. The 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.
Answer:
a. The equation for the given graph of the absolute value function in exploration 1 is y = |x|

b. y = √x

c. y = c

y = e^x

y = x³

y = x

y = 1/x

y = x²

Use graphing calculator to find the graph of the equation.

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.

Answer:

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

Question 2.
g(x) = x + 3
Answer:

Question 3.
h(x) = (x – 2)²

Answer:

Question 4.
n(x) = – | x |
Answer:

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

Question 5.
g(x) = 3x
Answer:

Question 6.
h(x) = \(\frac{3}{2}\)x²

Answer:

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

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
Answer:

Question 9.
d(x) = 3(x – 5)² – 1
Answer:

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?

Answer:

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.
Answer:
The function f(x) = x² is the parent function of f(x) = 2x² – 3.

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

Answer:

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.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.

Answer:

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.
Answer:

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:
Given,
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.

In Exercises 9–18, graph the function and its parent function. Then describe the transformation.

Question 9.
g(x) = x + 4
Answer:
Given equation is g(x) = x + 4

Question 10.
f(x) = x – 6
Answer:
f(x) = x – 6
y = x – 6
x = 0
y = 0 – 6 = -6
x = 1
y = 1 – 6 = -5
x = -1
y = -1 – 6 = -7
x = 2
y = 2 – 6 = -4
x = 3
y = 3 – 6 = -3
x = 4
y = 4 – 6 = -2
x = 5
y = 5 – 6 = -1
x = 6
y = 6 – 6 = 0
x = 7
y = 7 – 6 = 1
x = 8
y = 8 – 6 = 2

The graph is a linear function.

Question 11.
f(x) = x² – 1
Answer:
Given equation is f(x) = x² – 1

Question 12.
h(x) = (x+ 4)²
Answer:
Given equation is h(x) = (x+ 4)²

Question 13.
g(x) = | x – 5 |
Answer:

Question 14.
f(x) = 4 + | x |
Answer:

Question 15.
h(x) = -x²

Answer:
Given equation is h(x) = -x²

Question 16.
g(x) = -x
Answer:
Given equation is g(x) = -x
y = -x

Question 17.
f(x) = 3
Answer:
Given equation is f(x) = 3

Question 18.
f(x) = -2
Answer:
Given equation is f(x) = -2
The parent function is f(x) = 1
Find the graph of the given function y = -2

In Exercises 19–26, graph the function and its parent function. Then describe the transformation.

Question 19.
f(x) = \(\frac{1}{3}\)x
Answer:
Given equation is f(x) = \(\frac{1}{3}\)x
The parent function y = x
Find the graph of the given function f(x) = \(\frac{1}{3}\)x

Question 20.
g(x) = 4x
Answer:
Given equation is g(x) = 4x
The parent function if f(x) = x
Find the graph of the given function g(x) = 4x

Question 21.
f(x) = 2x²
Answer:
Given equation is f(x) = 2x²

Question 22.
h(x) = \(\frac{1}{3}\)x²
Answer:

Question 23.
h(x) = \(\frac{3}{4}\)x
Answer:
Given equation is h(x) = \(\frac{3}{4}\)x
The parent function y = x
Find the graph of the given function h(x) = \(\frac{3}{4}\)x

Question 24.
g(x) = \(\frac{4}{3}\)x
Answer:
Given equation is g(x) = \(\frac{4}{3}\)x
The parent function y = x
Find the graph of the given function g(x) = \(\frac{4}{3}\)x

Question 25.
h(x) = 3 | x |
Answer:
Given equation is h(x) = 3 | x |
The parent function y =|x|
Find the graph of the given function h(x) = 3 | x |

Question 26.
f(x) = \(\frac{1}{2}\) | x |
Answer:
Given equation is f(x) = \(\frac{1}{2}\) | x |
The parent function y =|x|
Find the graph of the given function f(x) = \(\frac{1}{2}\) | x |

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:
Given equation,
f(x) = 3x + 2
x = 0
y = 3(0) + 2 = 2
x = 1
y = 3(1) + 2 = 5
x = 2
y = 3(2) + 2 = 7
x = -1
y = 3(-1) + 2 = -3 + 2 = -1
x = -2
y = 3(-2) + 2 = -6 + 2 = -4

Question 28.
h(x) = -x + 5
Answer:
Given equation is h(x) = -x + 5
x = -5
y = -(-5) + 5 = 10
x = 0
y = 0 + 5 = 5
x = 5
y = -5 + 5 = 0

Question 29.
h(x) = -3 | x | – 1
Answer:
Given equation is h(x) = -3 | x | – 1
x = -3
y = -3|-3| – 1 = -9 – 1 = -10
x = -2
y = -3|-3| – 1 = -6 – 1 = -7
x = 0
y = -3|0| – 1 = -1
x = 3
y = -3|3| – 1 = -9 – 1 = -10
x = 2
y = -3|2| – 1 = -6 – 1 = -7

Question 30.
f(x) = \(\frac{3}{4}\) | x | + 1
Answer:
Given equation is f(x) = \(\frac{3}{4}\) | x | + 1
x = -5
y = \(\frac{3}{4}\) | x | + 1
y = \(\frac{3}{4}\) | -5 | + 1 = \(\frac{15}{4}\) + 1 = \(\frac{19}{4}\)
x = 0
y = \(\frac{3}{4}\) | 0 | + 1 = 1
x = -5
y = \(\frac{3}{4}\) | x | + 1
y = \(\frac{3}{4}\) | 5 | + 1 = \(\frac{15}{4}\) + 1 = \(\frac{19}{4}\)

Question 31.
g(x) = \(\frac{1}{2}\)x² – 6
Answer:
Given equation is g(x) = \(\frac{1}{2}\)x² – 6
The parent function is y = x²

Question 32.
f(x) = 4x² – 3
Answer:
Given equation is f(x) = 4x² – 3
The parent function is f(x) = x²

Question 33.
f(x) = -(x + 3)² + \(\frac{1}{4}\)
Answer:
Given equation is f(x) = -(x + 3)² + \(\frac{1}{4}\)
The parent function is f(x) = x²

Question 34.
g(x) = – | x – 1 | – \(\frac{1}{2}\)
Answer:
Given equation is g(x) = – | x – 1 | – \(\frac{1}{2}\)
The parent function is f(x) = |x|

ERROR ANALYSIS In Exercises 35 and 36, identify and correct the error in describing the transformation of the parent function.

Question 35.

Answer: The error is there is no vertical shrink of the parent quadratic function. The graph is a reflection in the x-axis followed by a vertical stretch of the parent quadratic function.

Question 36.

Answer: The graph is translated horizontally 3 units to the right.
So, the equation should be f(x) = |x – 3| not f(x) = |x + 3|

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: The function g is in the family of absolute value functions. The domain of the function is all real numbers and the range of the function is y ≥ -1.

Question 40.
h(x) = | x – 3 | + 2
Answer:
Given equation is h(x) = | x – 3 | + 2
x = -1
y = |-1 – 3| + 2 = 4 + 2 = 6
x = 0
y = |0 – 3| + 2 = 3 + 2 = 5
x = 1
y = |1 – 3| + 2 = 2 + 2 = 4

Question 41.
g(x) = 3x + 4
Answer:
Function g is in the family of linear functions.
Domain: All Real Numbers
Range:  All Real Numbers

Question 42.
f(x) = -4x + 11
Answer:
Given,
f(x) = -4x + 11
x = 0
y = -4(0) + 11 = 11
(0, 11)
x = 1
y = -4(1) + 11 = 7
(1, 7)
x = 2
y = -4(2) + 11 = -8 + 11 = -3
(2, -3)

Question 43.
f(x) = 5x² – 2
Answer:
Given function is f(x) = 5x² – 2

Question 44.
f(x) = -2x² + 6
Answer:
Given function is f(x) = -2x² + 6
x = -1
y = -2(-1)² + 6 = -2 + 6 = 4
(-1, 4)
x = 0
y = -2(0)² + 6 = 6
(0, 6)
x = 1
y = -2(1)² + 6 = -2 + 6 = 4
(1, 4)

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: The type of function that can model the data is an absolute value function because the data are linear and there are positive speeds for the positive and negative displacements.
The speed of the car 20 yards past the intersection is estimated to be 8 miles per hour.

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:
Given,
Graph the functions f(x) = | x – 4 | and g(x) = | x | – 4.

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.
Answer:
g represents a vertical shrink of f because the y-values of g are smaller than the y-values of f at the same x-values.
b. Describe how to transform the graph of f to obtain the graph of h.
Answer:
The graph of f would have to be reflected over the x-axis and then vertically stretched to obtain the graph of h.

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:
The distance traveled by the swimmer can be modeled by a linear function. The 10-meter head start is modeled by a vertical translation up to 10 units.

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:
The data can be modeled by an absolute value function. The x-intercept in this situation is when the battery of the computer dies.

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:
Given,

Question 56.
f(x) = | x | – 3; (-2, -5)
Answer:

Question 57.
f(x) = x – 3; (5, 2)
Answer:
Given,

Question 58.
f(x) = x – 4; (12, 8)
Answer:
Given,
f(x) = x – 4

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:
To find the x-intercept let y = 0, then solve for x.
y = x + 2
0 = x + 2
x + 2 = 0
x = 0 – 2
x = -2
To find the y-intercept let x = 0, then solve for y.
y = x+ 2
y = 0 + 2
y = 2
Therefore, the intercept is (0, 0) and the y-intercept is (-2, 2)

Question 61.
3x + y = 1
Answer:
Given,
3x + y = 1

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?

Answer:
The graphs of y = f (x) + k, y = f(x – h), and y = -f(x) are compared to the graph of the parent function by vertical shifts, horizontal shifts and reflections.
Vertical shifts: Let f(x) be the parent function and k be a positive number. To graph the function y = f (x) + k, we shift the graph of y = f(x) up k units by adding k to the y-coordinates of the points on the graph of f.
Horizontal shifts: Let f(x) be the parent function and h be a positive number. To graph the function y = f(x – h), we shift the graph of y = f(x) to the right h units by adding h to the x-coordinates of the points on the graph of f.
Reflections: Let f(x) be the parent function.
To graph the function y = -f(x), we reflect the graph of y = f(x) about the x-axis by multiplying the y-coordinates of the points on the graph of f by -1.

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 = [latex][\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
Answer:

Question 2.
f(x) = | x | – 3; translation 4 units to the right
Answer:

Question 3.
f(x) = – | x + 2 | – 1; reflection in the x-axis
Answer:

Question 4.
f(x) = [latex]\frac{1}{2}\)x+ 1; reflection in the y-axis
Answer:
Given function is f(x) = \(\frac{1}{2}\)x+ 1

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
Answer:
Given function is 4x + 2

Question 6.
f(x) = | x | – 3; vertical shrink by a factor of \(\frac{1}{3}\)
Answer:
Given function is f(x) = | x | – 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.
Answer:

Question 8.
WHAT IF? In Example 5, your revenue function is f(x) = 3x. How does this affect your profit for 100 downloads?
Answer:

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:
The function g(x) = | 5x |- 4 is a horizontal shrink of the function f(x) = | x | – 4.

Question 2.
WHICH ONE DOESN’T BELONG? Which transformation does not belong with the other three? Explain your reasoning.

Answer:
Horizontal shrink for a factor 1/5.

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:
A horizontal translation of 3 units right or a vertical translation of 3 units up will produce the function g from the function f.

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:
Given function is f(x) = | 2x | + 4

Question 20.
f(x) = | x+ 3 | ; horizontal stretch by a factor of 4
Answer:
Given function is f(x) = | x+ 3 |

Question 21.
f(x) = -2 | x – 4 | + 2

Answer:
Given function is f(x) = -2 | x – 4 | + 2

Question 22.
f(x) = 6 – x

Answer:
Given function is f(x) = 6 – x

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:
Given function is f(x) = x

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:
The transformation is a reflection in the x-axis.
We know that the area of the triangle is A = 1/2 × bh
A = 1/2 × 8 × 4 = 16 sq. units

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:
Given function is f(x) = x + 4
when x = 4
f(x) = x + 4
f(x) = 3 + 4 = 7
So, f(x) = 7

Question 47.
f(x) = 4x – 1; x = -1
Answer:
f(x) = 4x – 1
x = -1
f(-1) = 4(-1) – 1
= -4 – 1 = -5
So, f(-1) = -5

Question 48.
f(x) = -x + 3; x = 5
Answer:
Given function is f(x) = -x + 3
when x = 5
f(-5) = -(5) + 3
= -5 + 3 = -2

Question 49.
f(x) = -2x – 2; x = -1
Answer:
f(x) = -2x – 2
x = -1
f(-1) = -2(-1) – 2
= 2 – 2 = 0

Create a scatter plot of the data. (Skills Review Handbook)

Question 50.

Answer:
The coordinates are (8, 4), (10, 9), (11, 10), (12, 12) and (15, 12). Plot the points on the graph.

Question 51.

Answer:
The coordinates are (2, 22), (5, 13), (6, 15), (10, 12) and (13, 6).

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.

Answer:
The given function is linear.
The function g is shifted down and the slope of g is smaller than the slope of the parent linear function.
The domain of g and its parent function is all real numbers and the range of g and its parent function is all real numbers.

Question 2.

Answer:
The function g is a quadratic function. Function g is shifted left and g is narrower than the parent quadratic function.
The domain of g and its parent function is all real numbers and the range of g and its parent function is y > 0.

Question 3.

Answer:
Function g is an absolute value function. Function g is shifted left and up.
The domain of g and its parent function is all real numbers but the range of g is y > -2 and the range of its parent function is y > 0.

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
Answer:
Given,
f(x) = 3x
x = 0
y = 3(0) = 0
x = 1
y = 3(1) = 3
x = 2
y = 3(2) = 6
x = -1
y = 3(-1) = -3
x = -2
y = 3(-2) = -6
(0, 0), (1, 3), (2, 6), (-1, -3), (-2, -6).

Question 6.
f(x) = 2(x – 1)²
Answer:
f(x) = 2(x – 1)²
x = 0
f(x) = 2(0 – 1)² = 2
(0, 2)
x = 1
f(x) = 2(1 – 1)² = 0
x = 2
f(x) = 2(2 – 1)² = 2
x = -1
f(x) = 2(-1 – 1)² = 8
(0, 2), (1, 0), (2, 2) and (-1, 8).

Question 7.
f(x) = – | x + 2 | – 7
Answer:

Question 8.
f(x) = \(\frac{1}{4}\)x² + 1
Answer:

Question 9.
f(x) = –\(\frac{1}{2}\)x – 4
Answer:
Given,
f(x) = –\(\frac{1}{2}\)x – 4
x = 0
y = –\(\frac{1}{2}\)(0) – 4 = -4
x = 1
y = –\(\frac{1}{2}\)(1) – 4 = -4\(\frac{1}{2}\)
x = 2
y = –\(\frac{1}{2}\)(2) – 4 = -4 – 1 = -5
x = -1
y = –\(\frac{1}{2}\)(-1) – 4 = -3\(\frac{1}{2}\)
x = -2
y = –\(\frac{1}{2}\)(-2) – 4 = -4 + 1 = -3

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
Answer:
Given function is f(x) = 2x + 1
The vertex is (0, 1) of the original graph to move the vertex up by 3 units just add 3 to the y-intercept.
g(x) = 2x + 4

Question 11.
f(x) = -3 | x – 4 | ; vertical shrink by a factor of \(\frac{1}{2}\)
Answer:
Given the function is f(x) = -3 | x – 4 | to vertically shrink a function by a factor by c, multiply the whole function by c f(x) vertically shrunk by a factor of c would be cf(x)
so f(x) = -3 | x – 4 | vertically shrunk by a factor of 1/2 would be f(x) = (-3/2) |x – 4|

Question 12.
f(x) = 3 | x + 5 |; reflection in the x-axis
Answer:
Given,
f(x) = 3 | x + 5 |
The points reflected in the x-axis have opposite y-coordinates
f(x) = -y
-y = -3 |x + 5|
f(x) = -3 |x + 5|

Question 13.
f(x) = \(\frac{1}{3}\)x – \(\frac{2}{3}\) ; translation 4 units left
Answer:
Given,
f(x) = \(\frac{1}{3}\)x – \(\frac{2}{3}\)
= \(\frac{1}{3}\) (x – 2)
= \(\frac{1}{3}\) (x – 2 + 4)
= \(\frac{1}{3}\) (x + 2)
= \(\frac{1}{3}\)x + \(\frac{2}{3}\)
So, the translation 4 units left is f(x) = \(\frac{1}{3}\)x + \(\frac{2}{3}\)

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.
Answer:
Given,
f(x) =x
horizontal shrink by a factor of \(\frac{2}{3}\)
f(x) = \(\frac{3}{2}\)x
f(x) = \(\frac{3}{2}\)x – 2
g(x) =\(\frac{3}{2}\)x – 2
So, the translation 2 units down and a horizontal shrink by a factor of \(\frac{2}{3}\) is g(x) =\(\frac{3}{2}\)x – 2

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.
Answer:
Given,
f(x) =x
g(x) = f(x) – 9
It is also reflected about the y-axis.
g(x) = f(-x) – 9
So, the translation 9 units down followed by a reflection in the y-axis is g(x) = f(-x) – 9

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 |.
Answer:
Given,
f(x) = |x|
g(x) = bf(x)
Reflecting function over the x-axis
g(x) = -f(x)
Original function is f(x) = -|x|
Stretching by a factor of 4 means we have to multiply by 4.
g(x) = -4|x|
g(x) = -4x
translation 7 units down and 1 unit right of the graph of f(x) = | x |
g(x) = -4x – 7 -1
g(x) = -4x – 8
g(x) = -4(x + 2)
So, 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 is g(x) = -4(x + 2).

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 |.
Answer:
Given,
f(x) = |x|
Multiply output with 1/2 to vertically shrink function
g(x) = 1/2 |x|
Subtract 1 to the output of the function to translate 1 unit down and 2 units left.
g(x) = 1/2 |x| – 1 -2
g(x) = 1/2 |x| – 3
So, translation 1 unit down and 2 units left followed by a vertical shrink by a factor of \(\frac{1}{2}\) is g(x) = 1/2 |x| – 3.

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)

Answer:
From the given data in the above table,
2 months = 2300
5 months = 5750
6 months = 6900
9 months = 10,350
1 month = 2300/2 = 1150
12 months = 1150 × 12 = 13800
The mileage after 1 year is 13800.
The estimated mileage after 1 year is 14,000 miles.

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)
Answer:
Given,
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.
For senior citizen cost
= 1/2 (20x + 80) – 30
= 10x + 40 – 30
= 10x + 10
= 10(x + 1)
x = 3
= 10(3 + 1)
= 10(4)
= $40
Thus the total cost for a senior citizen to go camping for three days is $40.

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.

Answer:
Given,
A company purchases a copier for $12,000.
The spreadsheet shows how the copier depreciates over an 8-year period.
m = 12,000 – 10,750 = $1,250
b = 12,000
V = -1250t + 12,000
The slope shows that for every year that passes, the value depreciates by $1250.

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?
Answer:
One of the real-life situations is finding Variable costs.
Imagine that you are taking a taxi while on vacation. You know that the taxi service charges 9 rupees to pick your family up from your hotel and another 0.15 rupees per mile for the trip. Without knowing how many miles it will be to each destination, you can set up a linear equation that can be used to find the cost of any taxi trip you take on your trip. By using′ ′x′′ to represent the number of miles to your destination and ”y′′ to represent the cost of that taxi ride, the linear equation would be: y = 0.15x+9

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.
Answer:
The real-life example of straight-line depreciation is the decrease of the speed of the car by ten meters per second which was moving with an initial speed of a hundred meter per second till the speed reaches thirty meters per second.

Speed	Time (in second)
100	0
90	1
80	2
70	3
60	4
50	5
40	6
30	7

b. Write a function that models the depreciation.
Answer:
The real-life example of straight-line depreciation is the decrease of the speed of the car by ten meters per second which was moving with an initial speed of a hundred meter per second till the speed reaches thirty meters per second.
The function that represents the statement is y = 100 – 10x, where y is speed and x is time.

c. Sketch a graph of the function.
Answer:
y = 100 – 10x
x = 7
y = 100 – 10(7)
y = 100 – 70
y = 30
when x = 0
y = 100
when x = 4
y = 60

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:
The linear equation y = \(\frac{1}{2}\)x + 3 is written in slope-intercept form.

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:
The slope of the graph with a correlation coefficient of -0.98 is negative.

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:
Given that the slope of the line is 10, after 7 years the balance is $70.
The slope was correctly used in the situation, however, the intercept was not used correctly.
In this situation, the starting balance is $100, so after 7 years the balance is 100 + 70 = $170.

Question 12.

Answer:
Here the slope m = 3
The income is $3 per hour.

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:
Use the graphing calculator to find the equation of the line of best fit for the data.
Enter the data into two lists.
The line of best fit is y = 0.42x + 1.44
The correlation coefficient is r ≈ 0.61.
This represents a weak positive correlation.

Question 20.

Answer:

Question 21.

Answer:
Use the graphing calculator to find the equation of the line of best fit for the data.
Enter the data into two lists.
The line of best fit is y = -0.45x + 4.26
The correlation coefficient is r ≈ -0.67
This represents a weak negative correlation.

Question 22.

Answer:

Question 23.

Answer:
Use the graphing calculator to find the equation of the line of best fit for the data.
Enter the data into two lists.
The line of best fit is y = 0.61x + 0.10
The correlation coefficient is r ≈ 0.95.
This represents a strong positive correlation.

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?
Answer:
The points are (0, 30), (24, 0) are on the line.
We can assume the slope is k.
k = (30-0)/(0-24) = -5/4
And the slope represents the amount of money to be paid monthly

b. What is the domain and range of the function? What does each represent?
Answer:
Because x and y are linear functions, and
k = -5/4, (0, 30) is on the function.
y = -5/4 x + 30
And the domain is [0, 24] and the range is [0, 30] and the range represents the total money still to be paid.

c. How much do you still owe after making payments for 12 months?
Answer:
y = -5/4 x + 30
when x = 12 and y = 15
So, you still owe 15 hundred of dollars.

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:
The correlation coefficient is near to 0 and because of that, there line cannot help with the prediction.
Your friend is incorrect.
r = 0.3 is close to 0 than 1, the line of best fit will not make good predictions.

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:
As x value increases, y increases the value of z also decreases.
The correlation between x and z is negative.

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.
Answer:
It makes sense. There is a positive correlation between the average life expectancy and the country’s level of advancement and also between a country’s level of advancement and the number of personal computers per person.
Example: When compared to the USA, a lot of African countries don’t have nearly as many personal computers per person.

b. Is it reasonable to conclude that giving residents of a country personal computers will lengthen their lives? Explain.

Answer:
No, the number of personal computers per capita is only an indicator that a country is more advanced, not a way to prolong someone’s life expectancy.

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?
Answer:
A linear system is said to be consistent when it has at least one solution. A linear system is said to be 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

Answer:
2x – 3y = 3 —- × 2 ⇒ 4x – 6y = 6
-4x + 6y = 6

4x – 6y = 6
-4x + 6y = 6
0 ≠ 6
It has no solution.
So, the linear system is inconsistent.

b. 2x – 3y = 3
x + 2y = 5

Answer:
Given equations are
2x – 3y = 3
x + 2y = 5 —–eq.2 × 2
2x + 4y = 10

Solve 1 & 2
2x – 3y = 3
(-)2x + 4y = 10
-7y = -7
y = 1
x + 2 = 5
x = 5 – 2
x = 3
So, x = 3 and y = 1
It has two solutions.
So, the system is consistent

c. 2x – 3y = 3
-4x + 6y = 6

Answer:
Given equations
2x – 3y = 3
-4x + 6y = 6

2x – 3y = 3 —- × 2
-4x + 6y = 6

Solving 1 & 2,
4x – 6y = 6
-4x + 6y = 6
0
So, the linear system is inconsistent.

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

Answer:
Given equations
2x + y = 5
x – y = 1
Solving Eq. 1 & 2,
2x + y = 5
x – y = 1
3x = 6
x = 6/3
x = 2
Substitute the value of x in the eq. (2)
2 – y = 1
2 – 1 = y
y = 1
The linear system has one solution.

b. x+ 3y = 1
-x + 2y = 4

Answer:
Given equations are x+ 3y = 1
-x + 2y = 4

Solving 1 & 2,
x+ 3y = 1
-x + 2y = 4
5y = 5
y = 1
x + 3 = 1
x = 1 – 3
x = -2
The linear system has one solution.

c. x + y = 0
3x + 2y = 1

Answer:
Given equation,
Multiply eq. 1 by 2 to solving the equation
x + y = 0  — × 2 = 2x + 2y = 0
3x + 2y = 1

Solving 1 & 2
2x + 2y = 0
(-)3x + 2y = 1
-x = -1
x = 1
The linear system has one solution.

Communicate Your Answer

Question 3.
How can you determine the number of solutions of a linear system?
Answer:
A linear equation in two variables is an equation of the form ax + by + c = 0 where a, b, c ∈ R, a, and b ≠ 0. A system of linear equations usually has a single solution, but sometimes it can have no solution or an infinite solution. In the system of linear equations, we can find the number of solutions by comparing the coefficients of the variables of the given linear equations.

Question 4.
Suppose you were given a system of three linear equations in three variables. Explain how you would approach solving such a system.
Answer:

1. Solve one equation for one of its variables.
2. Substitute the expression from point 1 in the other two equations to obtain a linear system in two variables.
3. Solve the new linear system for both of its variables.
4. Substitute the values found in point 3 into one of the original equations and solve for the remaining variable.

Question 5.
Apply your strategy in Question 4 to solve the linear system.

Answer:
Given three equations
x + y + z = 1 — eq.1
x – y – z = 3 — eq. 2
-x – y – z = -1 —- eq. 3
Solving 1 & 2
x + y + z = 1
x – y – z = 3
2x = 4
x = 4/2
x = 2
Solving 1 & 3
x – y – z = 1
-x – y – z = -1
-2y -2z = 0
y + z = 0
y = -z

1.4 Lesson

Monitoring Progress

Question 1.
x – 2y + z = -11
3x + 2y – z = 7
-x + 2y + 4z = -9
Answer:

Question 2.
x + y – z = -1
4x + 4y – 4z = -2
3x + 2y + z = 0
Answer:

Question 3.
x + y + z = 8
x – y + z = 8
2x + y + 2z = 16
Answer:

Question 4.
In Example 3, describe the solutions of the system using an ordered triple in terms of y.
Answer:

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?
Answer:

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:
The solution of a system of three linear equations is expressed as an ordered triple.

Question 2.
WRITING Explain how you know when a linear system in three variables has infinitely many solutions.
Answer:
The system has infinitely many solutions when you have an identity such as 0 = 0.

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:
Given equations are
x + y – 2z = 5 — eq. 1
-x + 2y + z = 2 — eq. 2
2x + 3y – z = 9 — eq. 3

Question 4.
x + 4y – 6z = -1
2x – y + 2z = -7
-x + 2y – 4z = 5
Answer:
Given equations are
x + 4y – 6z = -1— (eq.1)
2x – y + 2z = -7 —- (eq. 2)
-x + 2y – 4z = 5 —-(eq. 3)
Solving eq.1 & eq.3
x + 4y – 6z = -1
-x + 2y – 4z = 5
6y – 10z = 4
3y – 5z = 2 — (eq. 4)
Solving (1) & (2)
x + 4y – 6z = -1 — × 2 ⇒ 2x + 6y – 12z = -2
2x – y + 2z = -7

2x + 6y – 12z = -2
2x – y + 2z = -7
–    +    –        +
7y – 14z = 5 —- (eq. 5)
Solving (2) & (3)
2x – y + 2z = -7 —- (2)
-x + 2y – 4z = 5 —-(3)—–×2 ⇒ -2x + 4y – 8z = 10

2x – y + 2z = -7
-2x + 4y – 8z = 10
3y – 5z = 3 — (eq. 6)
Solving 5 & 6
7y – 14z = 5 — × 3 ⇒ 21y – 42z = 15 — (eq. 7)
3y – 5z = 3 — × 7 ⇒ 21y – 35z = 21 — (eq. 8)

21y – 42z = 15
21y – 35z = 21
-7z = -6
z = 6/7 or 0.85

7y – 14z = 5
7y – 14(6/7) = 5
7y – 14(6) = 5
7y – 84 = 5
7y = 5 + 84
7y = 89
y = 89/7
y = 12.7

-x + 2y – 4z = 5
-x + 2(12.7) – 4(0.85) = 5
-x + 25.4 – 3.4 = 5
-x + 22 = 5
-x = 5 – 22
-x = -17
x = 17
The solution is x = 17, y = 12.7 and z = 0.85

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:
First, you have to eliminate one variable by using the substitution method. Then it becomes three linear equations in three variables, then you will know how to solve it.

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:
a. The given three circles in the graph has one common point. So, the system of equations has one solution.
b. Inside the small circle all the points are common. So, it has an infinite number of solutions.

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:
(3m + 1)²
It is in the form of (a + b)² = a² + b² + 2ab

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
Answer:

Question 2.
g(x) = | x | – 1
Answer:

Question 3.
h(x) = \(\frac{1}{2}\)x²

Answer:

Question 4.
h(x) = 4
Answer:

Question 5.
f(x) = -| x | – 3
Answer:

Question 6.
g(x) = -3(x + 3)²

Answer:

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
Answer:

Question 8.
f(x) = | x | ; vertical shrink by a factor of \(\frac{1}{2}\) followed by a translation 2 units up
Answer:

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?
Answer:
Given,
(x1, y1) = (10, 3.5)
(x2, y2) = (30, 10.5)
m = (10.5 – 3.5)/30 – 10
m = 7/20
y – 3.5 = 7/20(x – 10)
y – 3.5 = 7/20 x – 3.5
y = 7/20x
x = 45
y = 7/20 (45)
y = 15.75 miles
Therefore you can 15.75 miles in 45 minutes.

Question 12.
x + y + z = 3
-x + 3y + 2z = -8
x = 4z
Answer:

Question 13.
2x – 5y – z = 17
x + y + 3z = 19
-4x + 6y + z = -20
Answer:

Question 14.
x + y + z = 2
2x – 3y + z = 11
-3x + 2y – 2z = -13
Answer:

Question 15.
x + 4y – 2z = 3
x + 3y + 7z = 1
2x + 9y – 13z = 2
Answer:

Question 16.
x – y + 3z = 6
x – 2y = 5
2x – 2y + 5z = 9
Answer:

Question 17.
x + 2y = 4
x + y + z = 6
3x + 3y + 4z = 28
Answer:

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?

Answer:

Linear Functions Chapter Test

Write an equation of the line and interpret the slope and y-intercept.

Question 1.

Answer:

Question 2.

Answer:

Solve the system. Check your solution, if possible.

Question 3.
-2x + y + 4z = 5
x + 3y – z = 2
4x + y – 6z = 11
Answer:

Question 4.
y = \(\frac{1}{2}\)z
x + 2y + 5z = 2
3x + 6y – 3z = 9
Answer:

Question 5.
x – y + 5z = 3
2x + 3y – z = 2
-4x – y – 9z = -8
Answer:

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

Question 6.
Answer:

Question 8.
f(x) = 4
Answer:

Match the transformation of f(x) = x with its graph. Then write a rule for g.

Question 9.
g(x) = 2f(x) + 3

Answer:

Question 10.
g(x) = 3f(x) – 2
Answer:

Question 11.
g(x) = -2f(x) – 3
Answer:

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.

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.
Answer:

Linear Functions Cumulative Assessment

Question 1.
Describe the transformation of the graph of f(x) = 2x – 4 represented in each graph.

Answer:

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.
Answer:

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.

Answer:
Use the graphing calculator to find the equation.
From the graph, the correlation coefficient is r = -0.86
So, my friend is not correct, since the correlation coefficient is close to -1.

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
Answer:

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.
Answer:

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

Answer:

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
Answer:

Question 8.
Choose the correct equality or inequality symbol which completes the statement below about the linear functions f and g. Explain your reasoning.

Answer: