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# Pc12 sol c03_cp

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### Pc12 sol c03_cp

1. 1. 04_ch03a_pre-calculas12_wncp_solution.qxd 5/17/12 7:31 PM Page 26 Home Quit Checkpoint: Assess Your Understanding, pages 213–218 3.1 1. Multiple Choice The graph of y ϭ Ϫ3x3 ϩ 4 is translated 4 units right and 5 units down. What is an equation of the translation image? A. y = -3(x + 4)3 + 9 B. y = -3(x - 4)3 + 9 C. y = -3(x + 4)3 - 1 D. y = -3(x - 4)3 - 1 26 Chapter 3: Transforming Graphs of Functions—Checkpoint—Solutions DO NOT COPY. ©P
2. 2. 04_ch03a_pre-calculas12_wncp_solution.qxd 5/17/12 7:31 PM Page 27 Home Quit 2. Here is the graph of y = g(x). On the same grid, sketch the graph of each function below. State the domain and range of each function. a) y - 3 = g(x) Compare the equation to y ϭ g(x + 2) y ϭ g(x) y ؊ k ‫ ؍‬g(x): k ‫3 ؍‬ y So, mark some lattice points on y ‫ ؍‬g(x) and translate each point 3 units up. Both functions have domain: x ç ‫ޒ‬ 4 y + 1 ϭ g(x Ϫ 3) Both functions have 2 y Ϫ 3 ϭ g(x) range: y ç ‫ޒ‬ x Ϫ10 Ϫ4 Ϫ2 0 2 Ϫ2 Ϫ4 Ϫ6 Ϫ8 b) y = g(x + 2) Write y ‫ ؍‬g(x ؉ 2) as y ‫ ؍‬g( x ؊ (؊2)). Compare the equation to y ‫ ؍‬g(x ؊ h): h ‫2؊ ؍‬ Translate each point on the graph of y ‫ ؍‬g(x) 2 units left. The domain is: x ç ‫ޒ‬ The range is: y ç ‫ޒ‬ c) y + 1 = g(x - 3) Write y ؉ 1 ‫ ؍‬g(x ؊ 3) as y ؊ (؊1) ‫ ؍‬g(x ؊ 3). Compare the equation to y ؊ k ‫ ؍‬g(x ؊ h): h ‫ 3 ؍‬and k ‫1؊ ؍‬ Translate each point on the graph of y ‫ ؍‬g(x) 3 units right and 1 unit down. The domain is: x ç ‫ޒ‬ The range is: y ç ‫ޒ‬ ©P DO NOT COPY. Chapter 3: Transforming Graphs of Functions—Checkpoint—Solutions 27
3. 3. 04_ch03a_pre-calculas12_wncp_solution.qxd 5/17/12 7:31 PM Page 28 Home Quit 3. The graph of y = f(x) was translated to create each graph below. Write an equation of each graph in terms of the function f. y 8 6 y ϭ f(x) 4 x Ϫ2 0 2 4 6 8 a) y The graph of y ‫ ؍‬f(x) has a local maximum at (2, 9). 8 This graph has a local maximum y ϩ 3 ϭ f(x ϩ 5) 6 at (؊3, 6). So, the graph of y ‫ ؍‬f(x) was 4 translated 5 units left and 2 3 units down. The equation of the image x Ϫ8 Ϫ4 Ϫ2 0 2 graph has the form: y ؊ k ‫ ؍‬f(x ؊ h), where Ϫ2 h ‫ 5؊ ؍‬and k ‫3؊ ؍‬ So, an equation of the image graph is: y ؉ 3 ‫ ؍‬f(x ؉ 5) b) y The graph of y ‫ ؍‬f(x) has a local 10 maximum at (2, 9). This graph has a local maximum 8 at (5, 11). So, the graph of y ‫ ؍‬f(x) was 6 translated 3 units right and 4 2 units up. The equation of the image graph 2 y Ϫ 2 ϭ f(x Ϫ 3) has the form: y ؊ k ‫ ؍‬f(x ؊ h), x where h ‫ 3 ؍‬and k ‫2 ؍‬ 0 2 4 6 8 10 So, an equation of the image graph is: y ؊ 2 ‫ ؍‬f(x ؊ 3) 28 Chapter 3: Transforming Graphs of Functions—Checkpoint—Solutions DO NOT COPY. ©P
4. 4. 04_ch03a_pre-calculas12_wncp_solution.qxd 5/17/12 7:31 PM Page 29 Home Quit 3.2 4. Multiple Choice The graph of y = f(x) was y reflected in the x-axis. Which graph below 4 y ϭ f(x) is its reflection image? 2 x 0 2 A. y B. y x 4 Ϫ2 0 2 4 Ϫ2 x Ϫ2 0 C. y D. y x 4 0 4 Ϫ2 2 x Ϫ4 Ϫ4 Ϫ2 5. Here is the graph of y = k(x). On the same y grid, sketch and label the graph of each 4 function below, then state its domain and 2 range. y ϭ k(Ϫx) y ϭ k(x) x a) y = -k(x) Ϫ4 0 4 y ϭ Ϫk(x) The graph of y ‫؊ ؍‬k(x) is the image of the Ϫ2 graph of y ‫ ؍‬k(x) after a reflection in Ϫ4 the x-axis. Mark some lattice points on y ‫ ؍‬k(x), then reflect them in the x-axis. Mark these image points, then join them. Domain: x ç ‫ޒ‬ Range: y ◊ ؊2 b) y = k(-x) The graph of y ‫ ؍‬k(؊x) is the image of the graph of y ‫ ؍‬k(x) after a reflection in the y-axis. Mark some lattice points on y ‫ ؍‬k(x), then reflect them in the y-axis. Mark these image points, then join them. Domain: x ç ‫ޒ‬ Range: y » 2 ©P DO NOT COPY. Chapter 3: Transforming Graphs of Functions—Checkpoint—Solutions 29
5. 5. 04_ch03a_pre-calculas12_wncp_solution.qxd 5/17/12 7:31 PM Page 30 Home Quit 6. The graph of y = -x3 + 3x2 - x + 3 was reflected in the y-axis 10 y and its image is shown. What is an equation of the image? 8 6 When the graph of y ‫ ؍‬f(x) is reflected in the y-axis, the equation of its 4 image is y ‫ ؍‬f(؊x). 2 x So, an equation of the image is: Ϫ2 0 2 y ‫ ؍‬f(؊x) y ‫؊(؊ ؍‬x)3 ؉ 3(؊x)2 ؊ (؊x) ؉ 3 y ‫ ؍‬x3 ؉ 3x 2 ؉ x ؉ 3 3.3 7. Multiple Choice The point (Ϫ6, 2) lies on the graph of y = f(x). After vertical and horizontal stretches or compressions of the graph, the equation of the image is y = 3f(2x). Which point is the image of (Ϫ6, 2)? A. (Ϫ3, 6) B. (Ϫ12, 6) C. (Ϫ2, 4) D. (Ϫ18, 1) 8. Here is the graph of y = h(x). y 12 y ϭ 2h(4x) On the same grid, sketch the 8 y ϭ h(x) graph of each function below, then state its domain and range. 4 1 yϭ h(Ϫ2x) 1 3 a) y = 3h(-2x) x Ϫ4 2 4 6 8 Compare y ‫ ؍‬ah(bx) to Ϫ4 1 1 y ‫ ؍‬h( ؊2x): a ‫؍‬ and b ‫2؊ ؍‬ 3 3 So, the graph of y ‫ ؍‬h(x) is vertically compressed by a factor 1 1 of , horizontally compressed by a factor of , then reflected 3 2 in the y-axis. Use mental math and the transformation: (x, y) on y ‫ ؍‬h(x) corresponds to a , yb on y ‫ ؍‬h(؊2x), to x 1 1 ؊2 3 3 mark some image points, then join them. Domain: ؊4 ◊ x ◊ 2; range: ؊1 ◊ y ◊ 2 b) y = 2h(4x) Compare y ‫ ؍‬ah(bx) to y ‫2 ؍‬h(4x): a ‫ 2 ؍‬and b ‫4 ؍‬ So, the graph of y ‫ ؍‬h(x) is vertically stretched by a factor of 2, and 1 horizontally compressed by a factor of . Use mental math and the 4 transformation: (x, y) on y ‫ ؍‬h(x) corresponds to a , 2yb on y ‫2 ؍‬h(4x), x 4 to mark some image points, then join them. Domain: ؊1 ◊ x ◊ 2; range: ؊6 ◊ y ◊ 12 30 Chapter 3: Transforming Graphs of Functions—Checkpoint—Solutions DO NOT COPY. ©P
6. 6. 04_ch03a_pre-calculas12_wncp_solution.qxd 5/17/12 7:31 PM Page 31 Home Quit 9. The graph of y = g(x) is the image of 4 y y ϭ f(x) B A the graph of y = f(x) after a vertical x and/or horizontal stretch and/or 0 4 8 12 16 reflection. Corresponding points are Ϫ4 AЈ labelled. Write an equation of the image Ϫ8 y ϭ g(x) graph in terms of the function f. BЈ Point A(4, 2) on y ‫ ؍‬f(x) corresponds to point A؅(8, ؊6) on y ‫ ؍‬g(x). An equation for the image graph after a vertical or horizontal stretch or compression can be written in the form y ‫ ؍‬af(bx). A point (x, y) on y ‫ ؍‬f(x) corresponds to the point a , ayb on y ‫ ؍‬af(bx). x b The image of A(4, 2) is a , a(2)b , which is A؅(8, ؊6). 4 b 1 Equate the x-coordinates: b ‫؍‬ 2 Equate the y-coordinates: a ‫3؊ ؍‬ So, an equation of y ‫ ؍‬g(x) is: y ‫3؊ ؍‬f a xb 1 2 I used the coordinates of B and B؅, and mental math to verify the equation. ©P DO NOT COPY. Chapter 3: Transforming Graphs of Functions—Checkpoint—Solutions 31