Operations on Functions

Math
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Summary

Stretches and Shrinks

Summary Stretches and Shrinks

Stretches and Shrinks

We can also stretch and shrink the graph of a function. To stretch or shrink the graph in the y direction, multiply or divide the output by a constant. 2f (x) is stretched in the y direction by a factor of 2, and f (x) is shrunk in the y direction by a factor of 2 (or stretched by a factor of ). Here are the graphs of y = f (x), y = 2f (x), and y = x. Note that if (x, y1) is a point on the graph of f (x), (x, y2) is a point on the graph of 2f (x), and (x, y3) is a point on the graph of f (x), then y2 = 2y1 and y3 = y1. For example, (3, 2) is on the graph of f (x), (3, 4) is on the graph of 2f (x), and (3, 1) is on the graph of f (x).

Graphs of f (x), 2f (x), and f (x)

To stretch or shrink the graph in the x direction, divide or multiply the input by a constant. As in translating, when we change the input, the function changes to compensate. Thus, dividing the input by a constant stretches the function in the x direction, and multiplying the input by a constant shrinks the function in the x direction. f (x) is stretched in the x direction by a factor of 2, and f (2x) is shrunk in the x direction by a factor of 2 (or stretched by a factor of frac12). Here is a graph of y = f (x), y = f (x), and y = f (2x). Note that if (x1, y) is a point on the graph of f (x), (x2, y) is a point on the graph of f (x), and (x3, y) is a point on the graph of f (2x), then x2 = 2x1 and x3 = x1. For example, (- 2, 5) is on the graph of f (x), (- 4, 5) is on the graph of f (x), and (- 1, 5) is on the graph of f (2x).

Graphs of f (x), f (x), and f (2x)

We can understand the difference between altering inputs and altering outputs by observing the following:

If g(x) = 3f (x): For any given input, the output iof g is three times the output of f, so the graph is stretched vertically by a factor of 3.

If g(x) = f (3x): For any given output, the input of g is one-third the input of f, so the graph is shrunk horizontally by a factor of 3.