{"id":103000,"date":"2022-05-14T12:14:44","date_gmt":"2022-05-14T06:44:44","guid":{"rendered":"https:\/\/ccssanswers.com\/?p=103000"},"modified":"2022-05-10T14:22:56","modified_gmt":"2022-05-10T08:52:56","slug":"eureka-math-precalculus-module-2-lesson-7","status":"publish","type":"post","link":"https:\/\/ccssanswers.com\/eureka-math-precalculus-module-2-lesson-7\/","title":{"rendered":"Eureka Math Precalculus Module 2 Lesson 7 Answer Key"},"content":{"rendered":"

Engage NY Eureka Math Precalculus Module 2 Lesson 7 Answer Key<\/h2>\n

Eureka Math Precalculus Module 2 Lesson 7 Exercise Answer Key<\/h3>\n

Opening Exercise<\/p>\n

Consider the following matrices: A=\\(\\left[\\begin{array}{ll}
\n1 & 2 \\\\
\n2 & 4
\n\\end{array}\\right]\\), B=\\(\\left[\\begin{array}{ll}
\n2 & 0 \\\\
\n0 & 0
\n\\end{array}\\right]\\), and C=\\(\\left[\\begin{array}{cc}
\n2 & -2 \\\\
\n2 & 2
\n\\end{array}\\right]\\)
\na. Compute the following determinants.
\ni. det(A)
\nAnswer:
\ndet(A)=4-4=0<\/p>\n

ii. det(B)
\nAnswer:
\ndet(B)=0-0=0<\/p>\n

iii. det(C)
\nAnswer:
\ndet(C)=4-(-4)=8<\/p>\n

b. Sketch the image of the unit square after being transformed by each transformation.
\ni. LA<\/sub>\\(\\left(\\left[\\begin{array}{l}
\nx \\\\
\ny
\n\\end{array}\\right]\\right)\\)=\\(\\left[\\begin{array}{ll}
\n1 & 2 \\\\
\n2 & 4
\n\\end{array}\\right]\\left[\\begin{array}{l}
\nx \\\\
\ny
\n\\end{array}\\right]\\)
\nAnswer:
\n\"Eureka<\/p>\n

ii. LB<\/sub>\\(\\left(\\left[\\begin{array}{l}
\nx \\\\
\ny
\n\\end{array}\\right]\\right)\\)=\\(\\left[\\begin{array}{ll}
\n2 & 0 \\\\
\n0 & 0
\n\\end{array}\\right]\\left[\\begin{array}{l}
\nx \\\\
\ny
\n\\end{array}\\right]\\)
\nAnswer:
\n\"Eureka<\/p>\n

iii. LC<\/sub>\\(\\left(\\left[\\begin{array}{l}
\nx \\\\
\ny
\n\\end{array}\\right]\\right)\\)=\\(\\left[\\begin{array}{cc}
\n2 & -2 \\\\
\n2 & 2
\n\\end{array}\\right]\\left[\\begin{array}{l}
\nx \\\\
\ny
\n\\end{array}\\right]\\)
\nAnswer:
\n\"Eureka<\/p>\n

c. Find the area of each image of the unit square in Part b.
\nAnswer:
\nThe first two have no area because each image is a line segment. The third one is a square with sides of length \\(2 \\sqrt{2}\\), so its area is (\\(2 \\sqrt{2}\\))2<\/sup>=8.<\/p>\n

d. Explain the connection between the responses to Parts a and b.
\nAnswer:
\nThe determinant of the matrix A is the area of the image of the unit square under the transformation induced by A.<\/p>\n

Eureka Math Precalculus Module 2 Lesson 7 Exploratory Challenge Answer Key<\/h3>\n

Exploratory Challenge 1
\nFor each matrix A given below:
\ni. Plot the image of the unit cube under the transformation.
\nii. Find the volume of the image of the unit cube from part (i).
\niii. Does the transformation have an inverse? If so, what is the matrix that induces the inverse transformation?
\na. A=\\(\\left[\\begin{array}{lll}
\n2 & 0 & 0 \\\\
\n0 & 2 & 0 \\\\
\n0 & 0 & 2
\n\\end{array}\\right]\\)
\nAnswer:
\ni. \"Eureka
\nii. The image of the unit cube is a cube with sides of length 2. Thus, the volume of the image of the unit cube is 2 \u22c5 2 \u22c5 2=8.
\niii. This transformation has an inverse. To undo dilation by 2, we need to dilate by \\(\\frac{1}{2}\\). The matrix that represents the inverse transformation is B=\\(\\left[\\begin{array}{lll}
\n\\frac{1}{2} & 0 & 0 \\\\
\n0 & \\frac{1}{2} & 0 \\\\
\n0 & 0 & \\frac{1}{2}
\n\\end{array}\\right] .\\)<\/p>\n

b. A=\\(\\left[\\begin{array}{ccc}
\n4 & 0 & 0 \\\\
\n0 & 1 & 0 \\\\
\n0 & 0 & -2
\n\\end{array}\\right]\\)
\nAnswer:
\ni. \"Eureka
\nii. The image of the unit cube is a box with sides of length 4, 1, and 2. Thus, the volume of the image of the unit cube is 4\u22c51\u22c52=8.
\niii. This transformation is also invertible. To undo this transformation, we need to scale by \\(\\frac{1}{4}\\) in the
\nx-direction, by 1 in the y-direction, and by –\\(\\frac{1}{2}\\) in the z-direction. The matrix that will induce this inverse transformation is B=\\(\\left[\\begin{array}{ccc}
\n\\frac{1}{4} & 0 & 0 \\\\
\n0 & 1 & 0 \\\\
\n0 & 0 & -\\frac{1}{2}
\n\\end{array}\\right]\\).<\/p>\n

c. \\(\\left[\\begin{array}{ccc}
\n4 & 0 & 0 \\\\
\n0 & 0 & 0 \\\\
\n0 & 0 & -2
\n\\end{array}\\right]\\)
\nAnswer:
\ni. \"Eureka
\nii. The image of the unit cube is a rectangle with sides of length 2 and 4. Thus, the volume of the image of the unit cube is 4\u22c50\u22c52=0.
\niii. This transformation is not invertible since the image was collapsed from a cube onto a rectangle. Many points get sent to the same point on the rectangle, so we cannot undo the transformation in a clear way.<\/p>\n

d. Describe the geometric effect of a transformation \"Eureka for numbers a, b, and c. Describe when such a transformation is invertible.
\nAnswer:
\nThe transformation L scales the side of the cube along the x-axis by a, the side of the cube along the y-axis by b, and the side of the cube along the z-axis by c. If any of the numbers a, b, and c are zero, then the image is no longer a three-dimensional figure, so the transformation will not be invertible. Thus, the transformation is invertible only when a\u22600, b\u22600, and c\u22600.<\/p>\n

Exploratory Challenge 2.<\/p>\n

a. Make a prediction: What would be the geometric effect of the transformation
\n\"Eureka on the unit cube? Use the GeoGebra demo to test your conjecture.
\nAnswer:
\n\"Eureka
\nStudents should recognize that this transformation should have something to do with rotation. However, they most likely need to see the resulting image before identifying it as rotation by 45\u00b0 about the x-axis.<\/p>\n

b. For each geometric transformation below, find a matrix A so that the geometric effect of \"Eureka is the specified transformation.
\ni. Rotation by -45\u00b0 about the x-axis
\nAnswer:
\n\"Eureka<\/p>\n

ii. Rotation by 45\u00b0 about the y-axis
\nAnswer:
\n\"Eureka<\/p>\n

iii. Rotation by 45\u00b0 about the z-axis
\nAnswer:
\n\"Eureka<\/p>\n

iv. Rotation by 90\u00b0 about the x-axis
\nAnswer:
\n\"Eureka<\/p>\n

v. Rotation by 90\u00b0 about the y-axis
\nAnswer:
\n\"Eureka<\/p>\n

vi. Rotation by 90\u00b0 about the z-axis
\nAnswer:
\n\"Eureka<\/p>\n

vii. Rotation by \u03b8 about the x-axis
\nAnswer:
\n\"Eureka<\/p>\n

viii. Rotation by \u03b8 about the y-axis
\nAnswer:
\n\"Eureka<\/p>\n

viii. Rotation by \u03b8 about the z-axis
\nAnswer:
\n\"Eureka<\/p>\n

Exploratory Challenge 3.
\nDescribe the geometric effect of each transformation \"Eureka for the given matrices A. Be as specific as you can.
\na. A=\\(\\left[\\begin{array}{lll}
\n1 & 0 & 0 \\\\
\n0 & 0 & 1 \\\\
\n0 & 1 & 0
\n\\end{array}\\right]\\)
\n\"Eureka
\nAnswer:
\nThe images of the two cubes coincide; this transformation reflects across the plane through the points (1,0,0), (1,1,1), (0,1,1), and (0,0,0).<\/p>\n

b. A=\\(\\left[\\begin{array}{lll}
\n2 & 1 & 0 \\\\
\n2 & 1 & 0 \\\\
\n2 & 1 & 0
\n\\end{array}\\right]\\)
\n\"Eureka
\nAnswer:
\nThis transformation collapses the cube onto a line segment from (0,0,0) to (3,3,3).<\/p>\n

c. A=\\(\\left[\\begin{array}{ccc}
\n2 & 2 & 0 \\\\
\n-2 & 2 & 0 \\\\
\n0 & 0 & 1
\n\\end{array}\\right]\\)
\n\"Eureka
\nAnswer:
\nThis transformation stretches the cube in the x and y directions and then rotates by -45\u00b0.<\/p>\n

Eureka Math Precalculus Module 2 Lesson 7 Problem Set Answer Key<\/h3>\n

Problems 1 and 2 continue to develop the theory of linear transformations represented by matrix multiplication. Problem 3 can be done with or without access to the GeoGebra demo used in the lesson.<\/p>\n

Question 1.
\nSuppose that we have a linear transformation \"Eureka for some matrix A = \\(\\left[\\begin{array}{lll}
\na & d & g \\\\
\nb & e & h \\\\
\nc & f & i
\n\\end{array}\\right]\\).
\na. Evaluate L\\(\\left(\\left[\\begin{array}{l}
\n1 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right]\\right)\\). How does the result relate to the matrix A?
\nAnswer:
\nL\\(\\left(\\left[\\begin{array}{l}
\n1 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right]\\right)\\)=\\(\\left[\\begin{array}{l}
\na \\\\
\nb \\\\
\nc
\n\\end{array}\\right]\\) It is the first column of matrix A.<\/p>\n

b. Evaluate L\\(\\left(\\left[\\begin{array}{l}
\n0 \\\\
\n1 \\\\
\n0
\n\\end{array}\\right]\\right)\\). How does the result relate to the matrix A?
\nAnswer:
\nL\\(\\left(\\left[\\begin{array}{l}
\n0 \\\\
\n1 \\\\
\n0
\n\\end{array}\\right]\\right)\\)=\\(\\left[\\begin{array}{l}
\nd \\\\
\ne \\\\
\nf
\n\\end{array}\\right]\\) It is the second column of matrix A.<\/p>\n

c. Evaluate L\\(\\left(\\begin{array}{l}
\n\\left.\\left[\\begin{array}{l}
\n0 \\\\
\n0 \\\\
\n1
\n\\end{array}\\right]\\right)
\n\\end{array}\\right.\\). How does the result relate to the matrix A?
\nAnswer:
\nL\\(\\left(\\left[\\begin{array}{l}
\n0 \\\\
\n0 \\\\
\n1
\n\\end{array}\\right]\\right)\\)=\\(\\left[\\begin{array}{c}
\ng \\\\
\nh \\\\
\ni
\n\\end{array}\\right]\\) It is the third column of matrix A.<\/p>\n

d. James correctly said that if you know what a linear transformation does to the three points (1,0,0), (0,1,0), and (0,0,1), you can find the matrix of the transformation. Explain how you can find the matrix of the transformation given the image of these three points.
\nAnswer:
\nCreate a matrix with L(\\(\\left(\\left[\\begin{array}{l}
\n1 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right]\\right)\\)) in the first column, L(\\(\\left(\\left[\\begin{array}{l}
\n0 \\\\
\n1 \\\\
\n0
\n\\end{array}\\right]\\right)\\)) in the second column, and L(\\(\\left(\\left[\\begin{array}{l}
\n0 \\\\
\n1 \\\\
\n0
\n\\end{array}\\right]\\right)\\)) in the third column.<\/p>\n

Question 2.
\nUse the result from Problem 1(d) to answer the following questions.
\na. Suppose a transformation L:R3<\/sup>\u2192R3<\/sup> satisfies L(\\(\\left(\\left[\\begin{array}{l}
\n1 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right]\\right)\\))=[\\(\\left(\\left[\\begin{array}{l}
\n3 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right]\\right)\\), L(\\(\\left(\\left[\\begin{array}{l}
\n0 \\\\
\n1 \\\\
\n0
\n\\end{array}\\right]\\right)\\))=\\(\\left(\\left[\\begin{array}{l}
\n0 \\\\
\n3 \\\\
\n0
\n\\end{array}\\right]\\right)\\), and L(\\(\\left(\\left[\\begin{array}{l}
\n0 \\\\
\n0 \\\\
\n1
\n\\end{array}\\right]\\right)\\))=[\\(\\left(\\left[\\begin{array}{l}
\n0 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right]\\right)\\).<\/p>\n

i. What is the matrix A that represents the transformation L?
\nAnswer:
\nA=\\(\\left[\\begin{array}{lll}
\n3 & 0 & 0 \\\\
\n0 & 3 & 0 \\\\
\n0 & 0 & 3
\n\\end{array}\\right]\\)<\/p>\n

ii. What is the geometric effect of the transformation L?
\nAnswer:
\nThe transformation L has the geometric effect of dilation by a factor of 3.<\/p>\n

iii. Sketch the image of the unit cube after the transformation by L.
\nAnswer:
\n\"Eureka<\/p>\n

b. Suppose a transformation L:R3<\/sup>\u2192R3<\/sup> satisfies L(\\(\\left[\\begin{array}{l}
\n1 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right]\\))=\\(\\left[\\begin{array}{l}
\n1 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right]\\), L(\\(\\left[\\begin{array}{l}
\n0 \\\\
\n1 \\\\
\n0
\n\\end{array}\\right]\\))=[\\(\\left[\\begin{array}{l}
\n0 \\\\
\n1 \\\\
\n0
\n\\end{array}\\right]\\), and L(\\(\\left[\\begin{array}{l}
\n0 \\\\
\n0 \\\\
\n1
\n\\end{array}\\right]\\))=[\\(\\left[\\begin{array}{l}
\n0 \\\\
\n0 \\\\
\n-4
\n\\end{array}\\right]\\).<\/p>\n

i. What is the matrix A that represents the transformation L?
\nAnswer:
\nA=\\(\\left[\\begin{array}{ccc}
\n1 & 0 & 0 \\\\
\n0 & 1 & 0 \\\\
\n0 & 0 & -4
\n\\end{array}\\right]\\)<\/p>\n

ii. What is the geometric effect of the transformation L?
\nAnswer:
\nThe transformation L has the geometric effect of stretching by a factor of -4 in the z-direction.<\/p>\n

iii. Sketch the image of the unit cube after the transformation by L.
\nAnswer:
\n\"Eureka<\/p>\n

c. Suppose a transformation L:R3<\/sup>\u2192R3<\/sup> satisfies L\\(\\left(\\left[\\begin{array}{l}
\n1 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right]\\right)\\)=\\(\\left(\\left[\\begin{array}{l}
\n3 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right]\\right)\\), L\\(\\left(\\left[\\begin{array}{l}
\n0 \\\\
\n1 \\\\
\n0
\n\\end{array}\\right]\\right)\\)=\\(\\left[\\begin{array}{l}
\n0 \\\\
\n3 \\\\
\n0
\n\\end{array}\\right]\\), and L\\(\\left(\\begin{array}{l}
\n\\left.\\left[\\begin{array}{l}
\n0 \\\\
\n0 \\\\
\n1
\n\\end{array}\\right]\\right)
\n\\end{array}\\right.\\)=\\(\\left[\\begin{array}{l}
\n0 \\\\
\n0 \\\\
\n3
\n\\end{array}\\right]\\).<\/p>\n

i. What is the matrix A that represents the transformation L?
\nAnswer:
\nA=\\(\\left[\\begin{array}{ccc}
\n-2 & 0 & 0 \\\\
\n0 & 1 & 0 \\\\
\n0 & 0 & -2
\n\\end{array}\\right]\\)<\/p>\n

ii. What is the geometric effect of the transformation L?
\nAnswer:
\nThe transformation L has the geometric effect of rotating about the y-axis by 180\u00b0 and stretching by a factor of 2 in the x and z directions.<\/p>\n

iii. Sketch the image of the unit cube after transformation by L.
\nAnswer:
\n\"Eureka<\/p>\n

Question 3.
\nFind the matrix of the transformation that will produce the following images of the unit cube. Describe the geometric effect of the transformation.
\na. \"Eureka
\nAnswer:
\nThe geometric effect of this transformation is reflection across the xy-plane. The matrix that represents this transformation is A=\\(\\left[\\begin{array}{ccc}
\n1 & 0 & 0 \\\\
\n0 & 1 & 0 \\\\
\n0 & 0 & -1
\n\\end{array}\\right]\\).<\/p>\n

b. \"Eureka
\nAnswer:
\nThe geometric effect of this transformation is a stretch in the y-direction by a factor of 3. The matrix that represents this transformation is A=\\(\\left[\\begin{array}{lll}
\n1 & 0 & 0 \\\\
\n0 & 3 & 0 \\\\
\n0 & 0 & 1
\n\\end{array}\\right]\\).<\/p>\n

c. \"Eureka
\nAnswer:
\nThe geometric effect of this transformation is a stretch in the x- and z-directions by a factor of 2 and projection onto the xz-plane. The matrix that represents this transformation is A=\\(\\left[\\begin{array}{lll}
\n2 & 0 & 0 \\\\
\n0 & 0 & 0 \\\\
\n0 & 0 & 2
\n\\end{array}\\right]\\).<\/p>\n

d. \"Eureka
\nAnswer:
\nThe geometric effect of this transformation is a shear parallel to the xz-plane. The matrix that represents this transformation is A=\\(\\left[\\begin{array}{lll}
\n1 & 0 & 0 \\\\
\n0 & 1 & 0 \\\\
\n0 & 1 & 1
\n\\end{array}\\right]\\).<\/p>\n","protected":false},"excerpt":{"rendered":"

Engage NY Eureka Math Precalculus Module 2 Lesson 7 Answer Key Eureka Math Precalculus Module 2 Lesson 7 Exercise Answer Key Opening Exercise Consider the following matrices: A=, B=, and C= a. Compute the following determinants. i. det(A) Answer: det(A)=4-4=0 ii. det(B) Answer: det(B)=0-0=0 iii. det(C) Answer: det(C)=4-(-4)=8 b. Sketch the image of the unit … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/103000"}],"collection":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/comments?post=103000"}],"version-history":[{"count":1,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/103000\/revisions"}],"predecessor-version":[{"id":136153,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/103000\/revisions\/136153"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=103000"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=103000"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=103000"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}