116 CHAPTER 2 Functions and Their Graphs Based on the Exploration, we have the following result: Graphing Using Stretches and Compressions The graph of y f x( ) = is given in Figure 60 on the next page. Use this graph to find the graphs of (a) y f x 2 ( ) = (b) y f x3( ) = Solution EXAMPLE 3 (a) Because the 2 is “outside” the function f , the graph of y f x 2 ( ) = is obtained by multiplying each y -coordinate of y f x( ) = by 2. See Figure 61 on the next page. (b) Because the 3 is “inside” the function f , the graph of y f x3( ) = is obtained from the graph of y f x( ) = by multiplying each x -coordinate of y f x( ) = by 1 3 . See Figure 62 on the next page. Horizontal Compression or Stretch If the argument of a function y f x( ) = is multiplied by a positive number a , then the graph of the new function y f ax ( ) = is obtained by multiplying each x -coordinate of the graph of y f x( ) = by a 1 . • If a 1 > , a horizontal compression by a factor of a 1 results. • If a 0 1 < < , a horizontal stretch by a factor of a 1 results. In Words For ( ) = > y f ax a , 0, the factor a is “inside” the function, so it affects the x -coordinates. Multiply each x -coordinate on the graph of ( ) = y f x by a 1 . Figure 59 Horizontal stretch or compression Result You should have obtained the graphs and table in Figure 59. Look at the table. Note that 1, 1 ( ) , 4, 2 ( ) , and 9, 3 ( ) are points on the graph of Y x 1 = . Also, 0.5, 1 ( ) , 2, 2 ( ) , and 4.5, 3 ( ) are points on the graph of Y x2 2 = . For a given y -coordinate, the x -coordinate on the graph of Y2 is 1 2 of the x -coordinate on Y1 . We conclude that the graph of Y x2 2 = is obtained by multiplying the x -coordinate of each point on the graph of Y x 1 = by 1 2 . The graph of Y x2 2 = is the graph of Y x 1 = compressed horizontally. Look again at the table. Notice that 1, 1 ( ) , 4, 2 ( ) , and 9, 3 ( ) are points on the graph of Y x 1 = . Also notice that 2, 1 ( ) , 8, 2 ( ) , and 18, 3 ( ) are points on the graph of Y x 2 3 = . For a given y -coordinate, the x -coordinate on the graph of Y3 is 2 times the x -coordinate on Y1. We conclude that the graph of Y x 2 3 = is obtained by multiplying the x -coordinate of each point on the graph of Y x 1 = by 2. The graph of Y x 2 3 = is the graph of Y x 1 = stretched horizontally. Let’s look at an example.
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