Stretches and Shrinks Linear Functions

A horizontal shrink is a transformation that occurs when we multiply all of the x-coordinates (inputs) by the same factor a, where a>1. This causes the graph to shrink toward the y-axis.

(Note that we use the parabola rather than the line for illustration as it is easier to see the shrink from the red graph to the blue graph.)

*To create a horizontal shrink we multiply the x-value by a constant (a) greater than 1.

y = f(ax) where a>1. We say the graphy y is a horizontal shrink by a factor of 1/a.

*The y-intercept stays the same in a horizontal shrink.

The image shows f(x) = x+2 and f(2x). We say that the graph g(x) is a horizontal shrink by a factor of 1/2.

Note what happens to the x and y values after a horizontal shrink.

Note that the blue line is closer to the y-axis than the green line.

A horizontal stretch is a transformation where the graph stretches away from the y-axis.

In a stretch, 0 < a < 1.

*Note we illustrate with a parabola to see the stretch - f(x) is red and the stretch is blue. Notice that

a = 1/2.

*To create a horizontal stretch we multiply the x-value by a constant (a) where 0 < a < 1.

y = f(ax) where 0<a<1. We say the graph y is a horizontal stretch by a factor of a.

*The y-intercept stays the same in a horizontal shrink.

*The image shows f(x) = x+2 and f((1/2)x). We say that the graph g(x) is a horizontal stretch by a factor of 2.

A vertical shrink is a transformation that occurs when we multiply the y-coordinates (outputs) by the same factor a. When 0 < a <1, we have a vertical shrink.

y = a*f(x).

A vertical stretch occurs when we multipy the y-coordinate (output) by a factor a where a>1. A vertical stretch takes the graph further away from the x-axis.

In the case of a vertical stretch or shrink the x-intercept stays the same.

Sometimes we see that transformations are combined. Here is an example of the core concept.