Chapter 1 | Functions and Graphs
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example, consider the function f ( x ) = 2 x and evaluate f at x /2. Since f ( x /2) = x , the graph of f ( x ) = 2 x is the graph of y = x compressed horizontally. The graph of y = x /2 is a horizontal stretch of the graph of y = x ( Figure 1.26 ).
Figure 1.26 (a) If c >1, the graph of y = f ( cx ) is a horizontal compression of the graph of y = f ( x ). (b) If 0< c <1, the graph of y = f ( cx ) is a horizontal stretch of the graph of y = f ( x ).
We have explored what happens to the graph of a function f when we multiply f by a constant c >0 to get a new function cf ( x ). We have also discussed what happens to the graph of a function f when we multiply the independent variable x by c >0 to get a new function f ( cx ). However, we have not addressed what happens to the graph of the function if the constant c is negative. If we have a constant c <0, we can write c as a positive number multiplied by −1; but, what kind of transformation do we get when we multiply the function or its argument by −1? When we multiply all the outputs by −1, we get a reflection about the x -axis. When we multiply all inputs by −1, we get a reflection about the y -axis. For example, the graph of f ( x ) =−( x 3 +1) is the graph of y = ( x 3 +1) reflected about the x -axis. The graph of f ( x ) = (− x ) 3 +1 is the graph of y = x 3 +1 reflected about the y -axis ( Figure 1.27 ).
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