1. HORIZONTAL STRETCHES AND COMPRESSIONS
6.4
HORIZONTAL STRETCHES AND COMPRESSIONS

2. 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

3. 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)

4. 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.

5. 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)

6. 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.