# Operations on Functions

### 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, y 1) is a point on the graph of f (x) , (x, y 2) is a point on the graph of 2f (x) , and (x, y 3) is a point on the graph of f (x) , then y 2 = 2y 1 and y 3 = y 1 . 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 (x 1, y) is a point on the graph of f (x) , (x 2, y) is a point on the graph of f ( x) , and (x 3, y) is a point on the graph of f (2x) , then x 2 = 2x 1 and x 3 = x 1 . 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.

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