HORIZONTAL STRETCHES AND COMPRESSIONS

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HORIZONTAL STRETCHES AND COMPRESSIONS 6.4 HORIZONTAL STRETCHES AND COMPRESSIONS

Horizontally Compressed Example 1 The values and graph of the function f(x) are shown in blue. Make a table and a graph of the function g(x) = f(3x). Solution x f(x) g(x) -6 36 -2 -3 9 -1 1 -1/3 1/3 3 6 2

Horizontally Stretched Example 1 The values and graph of the function f(x) are shown in blue. Make a table and a graph of the function g(x) = f(½ x). Solution x f(x) g(x) -3 -6 -2 2 -4 -1 1 4 3 6 f(x) g(x) = f(½ x)

Formula for Horizontal Stretch or Compression If f is a function and k a positive constant, then the graph of y = f(k x) is the graph of f • Horizontally compressed by a factor of 1/k if k > 1, • Horizontally stretched by a factor of 1/k if k < 1. If k < 0, then the graph of y = f(kx) also involves a horizontal reflection about the y-axis.

Examples: Horizontal Stretch or Compression Example 3 Match the functions f(t) = et, g(t) = e0.5t, h(t) = e0.8t, j(t) = e2t with the graphs Solution Since the function j(t) = e2t climbs fastest of the four and g(t) = e0.5t climbs slowest, graph A must be j and graph D must be g. Similarly, graph B is f and graph C is h. A B C D j(t) f(t) h(t) g(t)

Ordering Horizontal and Vertical Transformations For nonzero constants A, B, h and k, the graph of the function y = A f(B (x − h)) + k is obtained by applying the transformations to the graph of f(x) in the following order: • Horizontal stretch/compression by a factor of 1/|B| • Horizontal shift by h units • Vertical stretch/compression by a factor of |A| • Vertical shift by k units If A < 0, follow the vertical stretch/compression by a reflection about the x-axis. If B < 0, follow the horizontal stretch/compression by a reflection about the y-axis.