122 CHAPTER 2 Functions and Their Graphs in the drop-down menu. Based on what you observe, conclude if the argument x of a function y f x( ) = is multiplied by a positive number a 0 1, < < then the graph of the new function y f ax ( ) = is obtained by multiplying each x-coordinate of y f x( ) = by . The new graph is a (horizontal/vertical) (stretch/compression) of the graph of y f x . ( ) = (c) If y f x( ) = is some function whose graph contains the point 12,5 ( ), the graph of y f x3( ) = would contain the point . Express your answer as an ordered pair. (d) If y f x( ) = is some function whose graph contains the point 3,2 ( ), the graph of ( ) = y f x 1 4 would contain the point . Express your answer as an ordered pair. 5. Suppose the graph of a function f is known. Then the graph of y f x 2 ( ) = − is obtained by a shift of the graph of f to the a distance of 2 units. 6. Suppose the graph of a function f is known. Then the graph of y f x ( ) = − is a reflection about the -axis of the graph of the function y f x . ( ) = 7. True or False The graph of y g x 1 3 ( ) = is the graph of y g x( ) = vertically stretched by a factor of 3. 8. True or False The graph of y f x( ) = − is the reflection about the x-axis of the graph of y f x . ( ) = 9. Multiple Choice Which function has a graph that is the graph of y x = shifted down 3 units? (a) y x 3 = + (b) y x 3 = − (c) y x 3 = + (d) y x 3 = − 10. Multiple Choice Which function has a graph that is the graph of y f x( ) = horizontally stretched by a factor of 4? (a) y f x4( ) = (b) ( ) = y f x 1 4 (c) y f x 4 ( ) = (d) y f x 1 4 ( ) = (c) If y f x( ) = is some function whose graph contains the point 2,4 ( ), the graph of y f x 3 ( ) = would contain the point . Express your answer as an ordered pair. (d) If y f x( ) = is some function whose graph contains the point 5,12 ( ), the graph of y f x 1 3 ( ) = would contain the point . Express your answer as an ordered pair. 4. Interactive Figure Exercise Exploring Horizontal Compressions and Stretches Open the “Horizontal Compressions and Stretches” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Use the drop-down menu to select the square root x ( ) function. The basic function f x x ( ) = is drawn in a dashed-blue line with four key points labeled. Set the slider labeled a to 1. Now, use the slider labeled a to slowly increase the value of a from 1 to 3. Carefully note the values of the x-coordinates on the graph of g (drawn in green) compared to the x-coordinates on the graph of f. For example, when a 2, = compare the x-coordinates on the two graphs. Also, notice the form of the function g x f ax . ( ) ( ) = Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if the argument x of a function y f x( ) = is multiplied by a positive number a 1, > then the graph of the new function y f ax ( ) = is obtained by multiplying each x-coordinate of y f x( ) = by . The new graph is a (horizontal/vertical) (stretch/compression) of the graph of y f x . ( ) = (b) Use the drop-down menu to select the square root x ( ) function. The basic function f x x ( ) = is drawn in a dashed-blue line with three key points labeled. Set the slider labeled a to 1. Now, use the slider labeled a to slowly decrease the value of a from 1 to 0.2. Carefully note the values of the x-coordinates on the graph of g (drawn in green) compared to the x-coordinates on the graph of f. For example, when a 0.5, = compare the x-coordinates on the two graphs. Also, notice the form of the function g x f ax . ( ) ( ) = Repeat this for other functions available Skill Building In Problems 11–22, match each graph to one of the following functions: A. y x 2 2 = + B. y x 2 2 = − + C. y x 2 = + D. y x 2 = − + E. y x 2 2 ( ) = − F. y x 2 2 ( ) = − + G. y x 2 = − H. y x 2 = − + I. y x2 2 = J. y x2 2 = − K. y x 2 = L. y x 2 = − 11. x y 3 3 23 12. x y 3 3 23 13. x y 1 3 23 14. x y 3 3 23
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