Transformations with Matrices Moved by Matrices

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1 Transformations with Matrices SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Marking the Text, Summarize/Paraphrase/Retell, Think/Pair/Share, Create Representations ACTIVITY 5.7 Instead of using real events, movie and television special effects are often computer generated. Often these effects are created through the use of simple images enhanced with transformations or complex images broken down into polygons which are then transformed. Computers use matrices to translate, reflect, and dilate images. A matrix is a rectangular array of numbers. Matrices are described using the number of rows first and then the number of columns. These are the dimensions of the matrix. For example, is a matrix. 7 Ordered pairs (x, y) can be expressed using a column matrix in the form x y. ACADEMIC VOCABULARY matrix MATH TERMS A column matrix is a matrix with only one column. 1. Write a column matrix for the ordered pair ( 3, 5) and state the dimensions. In a polygon, the ordered pairs that represent the vertices can all be placed into the columns of a matrix, resulting in a vertex matrix.. The quadrilateral ABCD has vertices A( 3, ), B(0, ), C(3, ) and D(0, 1). MATH TERMS A vertex matrix is a matrix containing the vertices of a polygon. a. Write a vertex matrix for ABCD. b. Label each column with the vertex it represents. c. What does each row in your vertex matrix represent? 3. State the type of polygon and the vertices represented in the following vertex matrix Unit 5 Coordinate Geometry and Transformations 395

2 ACTIVITY 5.7 Transformations with Matrices SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations Transformations of polygons can also be represented by matrices. B C D. Use pentagon ABCDE below to answer parts a h. a. Write the vertex matrix for pentagon ABCDE (preimage) A E b. Give the matrix that results from adding 3 to each x-coordinate and to each y-coordinate. c. State the vertices represented in this new matrix. d. Graph and label the resulting polygon (image) on the same set of axes as pentagon ABCDE. e. What type of transformation took place to create the image? To be added or subtracted, two matrices must have the same dimensions. 1 = f. What dimensions should any matrix added to your vertex matrix have? g. Write the matrix that was added to the vertex matrix of pentagon ABCDE to create the image. h. Write the matrix that would be added to the vertex matrix of pentagon ABCDE to create an image 5 units to the left and 3 units up from the preimage. 39 SpringBoard Mathematics with Meaning Geometry

3 Transformations with Matrices ACTIVITY 5.7 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations 5. Use GHI shown to the right to answer parts a g. a. Write the vertex matrix for GHI. H b. Multiply each element of the vertex matrix by. G I Write the new matrix. c. State the vertices represented in this new matrix. MATH TERMS An element of a matrix is a number in the matrix. d. Graph the image on the same set of axes as preimage GHI. e. What type of transformation took place to create the image? f. What is the relationship between the perimeter of the image and the perimeter of the preimage? g. What would need to be done to the vertex matrix of preimage GHI to create an image with a perimeter one half as large as that of GHI? Unit 5 Coordinate Geometry and Transformations 397

4 ACTIVITY 5.7 Transformations with Matrices SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Look for a Pattern. Use MNO to answer parts a g. y M N O x TECHNOLOGY a. Write the vertex matrix for MNO A graphing calculator can be used to multiply matrices. b. Multiply the matrix 1 resulting matrix by the vertex matrix. Write the c. Which numbers in the product matrix stayed the same? d. Which numbers in the product matrix changed? e. Graph the image on the same set of axes as preimage MNO. f. What type of transformation took place to create the image? Be specific. 39 SpringBoard Mathematics with Meaning Geometry

5 Transformations with Matrices ACTIVITY 5.7 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Look for a Pattern, Quickwrite, Self/Peer Revision 7. Use MNO to answer parts a e. y M N O x a. Multiply the matrix resulting matrix by the vertex matrix. Write the 1 b. Which numbers in the product matrix stayed the same? c. Which numbers in the product matrix changed? d. Graph the image on the same set of axes as preimage MNO. e. What type of transformation took place to create the image? Be specific.. Compare and contrast the matrices used to transform the vertex matrices and the product matrices in items b and 7a. Explain how the placement of the 1 affected the transformation of each of the preimages. Unit 5 Coordinate Geometry and Transformations 399

6 ACTIVITY 5.7 Transformations with Matrices SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Create Representations, Look for a Pattern, Identify a Subtask 9. Use PQR to answer parts a d. y Q P R x WRITING MATH Recall that prime and double prime notation indicate corresponding parts of congruent or similar figures. In PQR and P'Q'R', P' corresponds to P, Q' corresponds to Q and R' corresponds to R. In P'Q'R' and P"Q"R", P" corresponds to P', Q" corresponds to Q' and R" corresponds to R'. MATH TERMS The combination of two reflections, one over the x-axis and one over the y-axis is also referred to as a reflection over the origin. a. Write the vertex matrix for PQR. b. Use matrix multiplication to reflect PQR in the x-axis. State the matrix you used for this reflection and graph the image. Label the image P'Q'R'. c. Use matrix multiplication to reflect P'Q'R' in the y-axis. State the matrix you used for this reflection and graph the image. Label the image P"Q''R''. d. The transformation from PQR to P''Q''R'' can be accomplished in one step by multiplying the preimage by a single matrix. Write the matrix you think should be used. Justify your answer. 00 SpringBoard Mathematics with Meaning Geometry

7 Transformations with Matrices ACTIVITY 5.7 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share. Multiply each of the following matrices by the vertex matrix of EFG, graph the image, and describe the direction and angle of rotation about the origin that results from each. a y E F b c G x 11. Use your answers from Items to determine what matrix operation is needed to perform each of the following transformations on a vertex matrix for a quadrilateral. Be specific. a. dilation by a scale factor of 5 b. reflection in the origin c. rotation 90 clockwise about the origin d. reflection over the x-axis Unit 5 Coordinate Geometry and Transformations 01

8 ACTIVITY 5.7 Transformations with Matrices SUGGESTED LEARNING STRATEGIES: Think/Pair/Share e. rotation 10 counterclockwise about the origin f. translation right 5 units and down units g. rotation 70 clockwise about the origin h. reflection over the y-axis CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 1. A game designer is creating a graphic by transforming various polygons. He begins with GHI having G(-3, 5), H(, ), and I(7, 5). a. Use a matrix to find the coordinates of the image after he rotates the triangle 90 clockwise about the origin. Show the matrices you used.. MATHEMATICAL REFLECTION a. Write a matrix that could be used to reflect a polygon over the line y = x. Explain your answer. b. Write a matrix that could be used to reflect a polygon over the line y = x. Explain your answer. b. What would be the coordinates of the image if he decided to enlarge the preimage to 3 times its initial size and then translate it up units and 5 units left? Show the matrices that you used to find your answer. 0 SpringBoard Mathematics with Meaning Geometry

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