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Name June 3/4, 2015 Math 4 notes and problems Transformation rules and matrices page 1 Transformation rules and matrices Objective: Write algebraic rules representing transformations, and express those rules using matrix multiplication. Transformation rules Previously we have studied transformations in the xy-plane, such as translations, reflections, rotations, and dilations. In Chapter 1, we learned how to modify a function formula in order to transform the function’s graph. But what we haven’t addressed yet is how to give an algebraic description of the transformation itself. A transformation rule is a description of a transformation that tells how to get the output point for any given input point. The following notation is typically used: (x, y) stands for the input point. (x′, y′) stands for the output point, also called the image. One way to give a transformation rule is to write a pair of equations. Here is an example: x′ = x + 3 y′ = y – 1 These equations say that given any point (x, y), the image point (x′, y′) is found by adding 3 to the x-coordinate and subtracting 1 from the y-coordinate. (How would you describe this transformation geometrically?) Many geometrically interesting transformations have fairly simple equations. In fact, all translations, reflections, rotations, and dilations can be described using equations of the form x′ = ax + by + h y′ = cx + dy + k An important fact about transformation equations having this form is that the image of a line segment is always a line segment. Therefore, if you want to see the effect of a transformation on a polygon, you just need to find the images of the vertices. Problems 1. For each transformation rule below, geometrically describe the transformation. It may help to try some examples of input-output calculations. a. x′ = –x y′ = y b. x′ = 2x y′ = y + 3 c. x′ = y y′ = x 2. Answer the following questions about the transformation x′ = x + 2y, y′ = 3x – y. a. Find the image of the point (4, 5). b. Find a point (x, y) whose image is the point (5, 8). Hint: Set x′ = 5 and y′ = 8 to get a system of linear equations that can be solved. Name June 3/4, 2015 Math 4 notes and problems Transformation rules and matrices page 2 3. Apply each of the following transformations to the given “house” figure. (You can do this by transforming points A through F, then connecting the dots.) Then, describe the transformation using geometric words. a. x′ = x – 7, y′ = y + 3 b. x′ = –x, y′ = y Name June 3/4, 2015 c. x′ = 2x, y′ = –y d. x′ = –y, y′ = x Math 4 notes and problems Transformation rules and matrices page 3 Name June 3/4, 2015 Math 4 notes and problems Transformation rules and matrices page 4 Name June 3/4, 2015 Math 4 notes and problems Transformation rules and matrices page 5 Transformation matrices A linear transformation is a transformation whose rule has the form x′ = ax + by, y′ = cx + dy. Such a rule can always be rewritten as a matrix multiplication equation: ⎡a b ⎤ ⎡ x ⎤ ⎡ x ʹ′⎤ ⎢ c d ⎥ ⋅ ⎢ y ⎥ = ⎢ y ʹ′⎥ . ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ The 2-by-2 matrix is called the transformation matrix for this transformation. The useful thing about rewriting a transformation rule using a transformation matrix is that image points can now be calculated using matrix multiplication. Example Consider the transformation from problem 2: x′ = x + 2y, y′ = 3x – y. The rule can be written as ⎡1 2 ⎤ ⎡ x ⎤ ⎡ x ʹ′ ⎤ ⎢3 − 1⎥ ⋅ ⎢ y ⎥ = ⎢ y ʹ′⎥ . ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ You can find the image of point (4, 5) by doing this matrix multiplication: ⎡1 2 ⎤ ⎡4⎤ ⎢3 − 1⎥ ⋅ ⎢5⎥ = ⎣ ⎦ ⎣ ⎦ It gets better: you can calculate the images of all the vertex points of the “house” figure using one big matrix multiplication! ⎡1 2 ⎤ ⎡2 2 4 5 6 6⎤ ⎢3 − 1⎥ ⋅ ⎢1 3 3 4 3 1⎥ = ⎣ ⎦ ⎣ ⎦ Name June 3/4, 2015 Math 4 notes and problems Transformation rules and matrices page 6 Problems 4. Here is a transformation rule written using a matrix: ⎡− 2 0⎤ ⎡ x ⎤ ⎡ x ʹ′ ⎤ ⎢ 0 1⎥ ⋅ ⎢ y ⎥ = ⎢ y ʹ′⎥ . ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ a. Calculate the image of the point (5, 4). b. Calculate the image of the point (a, b). c. Calculate and draw the image of this figure. You can use your calculator to do the matrix multiplication. d. Describe the transformation using geometric words. Name June 3/4, 2015 Math 4 notes and problems Transformation rules and matrices page 7 5. Here is a transformation rule written using a matrix: ⎡0 1⎤ ⎡ x ⎤ ⎡ xʹ′ ⎤ ⎢1 0⎥ ⋅ ⎢ y ⎥ = ⎢ yʹ′⎥ . ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ a. Calculate the image of the point (5, 4). b. Calculate the image of the point (a, b). c. Calculate and draw the image of this figure. You can use your calculator to do the matrix multiplication. d. Describe the transformation using geometric words. Name June 3/4, 2015 Math 4 notes and problems Transformation rules and matrices page 8 6. For each transformation, write a transformation rule as a pair of equations. a. reflection across the y-axis x′ = y′ = b. translation down by 5 and left by 1 x′ = y′ = c. stretching horizontally by a factor of 2 and shrinking vertically by a factor of 1 2 x′ = y′ = 7. For each transformation, write a transformation rule as a matrix multiplication equation. a. stretching by a factor of 3 both horizontally and vertically b. reflecting across the x-axis and stretching vertically by a factor of 5 c. reflecting across the x-axis and stretching horizontally by a factor of 5 8. The goal of this problem is to figure out what transformation is represented by the matrix ⎡ 0 − 1⎤ ⎢− 1 0 ⎥ ⎣ ⎦ a. On graph paper, draw a figure made up by you that is comparable to my “house.” (Choose a figure that has about 6-to-8 vertex points.) b. Apply the transformation to your figure. (You may use your calculator for the matrix multiplication.) Draw the image figure. c. Describe the transformation geometrically.