The transformation of a function allows us to make modifications to its graph. One of these transformations is the stretching and compression of functions. We can compress or stretch the function on the *x*-axis when we have *f*(*ax*) and we can compress or stretch the function on the *y*-axis when we have *af*(*x*) where *a* is a constant.

In this article, we will learn how to stretch or compress a function both with respect to the *x*-axis and with respect to the *y*-axis.

## Stretches and compressions of a function with respect to the *x*-axis and the *y*-axis

Stretches and compressions are transformations that are produced when the *x* or *y* values of the original function are multiplied by a constant value.

To understand the stretches and compressions with respect to the *x*-axis and the *y*-axis, we are going to use the function $latex f(x)=x+1$. By graphing this function, we get the following line:

To produce stretches and compressions, we are going to multiply by a constant either the *x*-values or *y*-values of the function $latex f(x)$. Therefore, we have (i) $latex f(2x)$ and (ii) $latex 2f(x)$. Simplifying, we have:

(i) $latex f(2x)=(2x)+1~$ and (ii) $latex 2f(x)=2(x+1)$

(i) $latex f(2x)=2x+1~$ and (ii) $latex 2f(x)=2x+2$

When we plot functions (i) and (ii) together with the original function $latex f(x)$, we have:

In transformation (i), the graph of $latex f(x)$ has been stretched about the *x*-axis by a factor of $latex \frac{1}{2}$ (the function was halved)

In transformation (ii), the graph of $latex f(x)$ has been stretched about the *y*-axis by a factor of 2.

In short, we have:

- The transformation $latex f(ax)$ results in a stretch about the
*x*-axis by a factor of $latex \frac{1}{a}$. - The transformation $latex af(x)$ results in a stretch about the
*y*-axis by a factor of $latex a$. - If the stretch factor is between 0 and 1, the transformation is a compression of the graph.

## Examples of stretching and compressing functions

The following examples use the function stretch and compress transformations. Each example has a detailed solution, but try to solve the problems yourself first.

**EXAMPLE 1**

Obtain the graph of the function $latex f(x)=x+2$, and then graph the function $latex g(x)=3f(x)$.

##### Solution

The graph of the function $latex f(x)=x+2$ is a line that intersects the *y*-axis at (0, 2) and the *x*-axis at (-2, 0):

The function *g* is given by $latex g(x)=3f(x)=3x+6$. The graph of this function is equal to the graph of *f* stretched by a factor of 3 about the *y*-axis.

.

**EXAMPLE **2

**EXAMPLE**2

What is the difference between the graphs of $latex f(x)=\cos(x)$ and the graphs of $latex g(x)=\cos(2x)$ and $latex h(x)=\cos( \frac{1}{2}x)$?

##### Solution

When we apply the transformation $latex g(x)=f(ax)$, where *a* is a constant, we produce a stretch or compression about the *x*-axis.

Therefore, we can look at the graph of the standard cosine function $latex f(x)=\cos(x)$ along with the other two functions:

We see that the graph of $latex g(x)=\cos(2x)$ is halved, i.e. the stretch factor is $latex \frac{1}{a}=\frac{1}{2 }$.

In the case of the graph of $latex h(x)=\cos(\frac{1}{2}x)$, the function is stretched. The stretch factor is $latex \frac{1}{a}=\frac{1}{\frac{1}{2}}=2$.

**EXAMPLE **3

**EXAMPLE**3

Graph the functions $latex g(x)=2|x|$ and $latex h(x)=0.5|x|$.

##### Solution

In this case, we have the absolute value function, which in its base form, $latex f(x)=|x|$, has the following graph:

Now, the functions *g* and *h* are obtained by applying stretches with respect to the *y*-axis. That is, we have $latex g(x)=2f(x)$ and $latex h(x)=0.5f(x)$.

We see that the function *h* is stretched by a factor of 2 and the function *g* by a factor of 0.5 (equals a compression by half).

**EXAMPLE **4

**EXAMPLE**4

What changes do we need to make to the function $latex f(x)=3x^2+6x$ if we want to stretch it by a factor of 3 about the *x*-axis?

##### Solution

To stretch a function by a factor of 3 about the *x*-axis, we have to apply the transformation $latex g(x)=f(\frac{1}{3}x)$.

This means that we have to replace the variable *x* in *f* with $latex \frac{1}{3}x$. Therefore, we have:

$latex g(x)=f(\frac{1}{3}x)$

$latex g(x)=3(\frac{1}{3}x)^2+6(\frac{1}{3}x)$

$latex g(x)=3(\frac{1}{9})x^2+2x$

$latex g(x)=\frac{1}{3}x^2+2x$

## Stretching and compression of functions – Practice problems

Apply everything you have learned about stretching and compressing functions to solve the following practice problems.

## See also

Interested in learning more about function transformations? Take a look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**