SECTION 2.5 Graphing Techniques: Transformations 121 Concepts and Vocabulary 2.5 Assess Your Understanding 1. Interactive Figure Exercise Exploring Vertical Transformations Open the “Horizontal and Vertical Transformations” 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 absolute value x ( ) function. The basic function f x x ( ) = is drawn in a dashed-blue line with three key points labeled. Now, use the slider labeled k to slowly increase the value of k from 0 to 3.As you do this, notice the form of the function g x f x h k ( ) ( ) = − + labeled in green. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if a positive real number k is added to the outputs of a function y f x , ( ) = the graph of the new function g x f x k ( ) ( ) = + is the graph of f shifted (horizontally/vertically) (up/down/left/right) k units. (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. Now, use the slider labeled k to slowly decrease the value of k from 0 to 3. − As you do this, notice the form of the function g x f x h k ( ) ( ) = − + labeled in green. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if a positive real number k is subtracted from the outputs of a function y f x , ( ) = the graph of the new function g x f x k ( ) ( ) = − is the graph of f shifted (horizontally/vertically) (up/down/left/right) k units. (c) If y f x( ) = is some function whose graph contains the point 2,4 ( ) , the graph of y f x 2 ( ) = + 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 7 ( ) = − would contain the point . Express your answer as an ordered pair. 2. Interactive Figure Exercise Exploring Horizontal Transformations Open the “Horizontal and Vertical Transformations” 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 absolute value x ( ) function. The basic function f x x ( ) = is drawn in a dashed-blue line with three key points labeled. Now, use the slider labeled h to slowly increase the value of h from 0 to 4. As you do this, notice the form of the function g x f x h k ( ) ( ) = − + labeled in green. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if the argument x of a function is replaced by x h h , 0, − > the graph of the new function g x f x h ( ) ( ) = − is the graph of f shifted (horizontally/vertically) (up/down/left/right) h units. (b) Use the drop-down menu to select the absolute value x ( ) function. The basic function f x x ( ) = is drawn in blue with three key points labeled. Now, use the slider labeled h to slowly decrease the value of h from 0 to 4. − As you do this, notice the form of the function g x f x h k ( ) ( ) = − + labeled in green. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if the argument x of a function is replaced by x h h , 0, + > the graph of the new function g x f x h ( ) ( ) = + is the graph of f shifted (horizontally/vertically) (up/down/left/right) h units. (c) If y f x( ) = is some function whose graph contains the point 3,4 ( ) , 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 3,2 ( ) , the graph of y f x 4 ( ) = + would contain the point . Express your answer as an ordered pair. 3. Interactive Figure Exercise Exploring Vertical Compressions and Stretches Open the “Vertical 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 absolute value 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 increase the value of a from 1 to 3.As you do this, notice the form of the function g x af x ( ) ( ) = and the behavior of the graph of the function g x af x ( ) ( ) = shown in green. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude when the right side of a function y f x( ) = is multiplied by a positive number a 1, > the graph of the new function is obtained by multiplying each y -coordinate on the graph of y f x( ) = by . The new graph is a (horizontally/vertically) (stretched/compressed) version of the graph of y f x . ( ) = (b) Use the drop-down menu to select the absolute value 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. As you do this, notice the form of the function g x af x ( ) ( ) = and the behavior of the graph of the function g x af x ( ) ( ) = shown in green. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude when the right side of a function y f x( ) = is multiplied by a positive number a 0 1, < < the graph of the new function is obtained by multiplying each y -coordinate on the graph of y f x( ) = by . The new graph is a (horizontally/vertically) (stretched/compressed) version of the graph of y f x . ( ) = 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure (continued)
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