Stretch or Compress Function Vertically: cf(x)

Stretch or Compress Vertically

Another transformation that can be applied to a function is stretching or compressing a function vertically. $\\$ We can multiply a function by a positive constant $c$ to stretch and compress a function vertically. $\\$ Given the graph of $f(x)$, $cf(x)$ will be the graph of $f(x)$ stretched vertically if the value of $c$ is greater than $1$ ($c>1$). $cf(x)$ will be the graph of $f(x)$ compressed vertically if the value of $c$ is greater than $0$ but less than $1$ ($0<c<1$).

Example 1

$g(x)=3(x^2)$ $\\$ The graph $g(x)=3(x^2)$ will be a vertical stretch of $f(x)=x^2$ by $3$ units.$\\$ The red graph is the graph of $f(x)=x^2$ and the blue graph is the graph of $g(x)=3(x^2)$.

Example 2

Graph of $g(x)=\dfrac{1}{3}(x^2)$ $\\$ The graph $g(x)=\dfrac{1}{3}(x^2)$ will be a vertical compress of $f(x)=x^2$ by $3$ units. The red graph is the graph of $f(x)=x^2$ and the blue graph is the graph of $g(x)=\dfrac{1}{3}(x^2)$.

(Video) Vertical Stretch or Compression of a Quadratic Function

by Math and Stats Help

In this video by Math and Stats Help, you'll learn how to determine whether a quadratic function has been vertically stretched or compressed.

Summary

This video mainly talks about quadratics, but the technique can be applied to any function.

If a number value is multiplied outside of the function, it creates a vertical stretch or vertical compression.

Vertical stretch: $a \cdot f(x)$

If $a>1$, the graph is stretched vertically by a factor of a $a$.

Vertical compression: $a \cdot f(x)$

If $0<a<1$, the graph is compressed vertically by a factor of $a$. We can think of vertical compressions as being compressed toward the $x$-axis.

Example

The first example is for a vertical stretch.

$$f(x) = 2 \cdot x^2$$

We know this function has a vertical stretch because $2>1$. The graph of the parent function is in purple, $f(x)=x^2$, and the graph of the transformed function, $f(x)=2 \cdot x^2$, is in green.

Example

This second example is for a vertical compression.

$$f(x) = \frac12 x^2$$

We know that this is a compression because $0< \frac12 < 1$. The graph of the parent function is in purple, $f(x)=x^2$, and the graph of the transformed function, $f(x)=\frac12 x^2$, is in green.

Vertical Dilation

In context

In our series of function transformations, we learned about vertical translations, horizontal translations, vertical reflections, and horizontal reflections.

Next in our series of function transformations is the vertical dilation!

We can think of a vertical dilation as pushing or pulling a function toward or away from the $x$-axis. At first, you may be thinking that the $x$-axis is the horizontal axis, so that doesn't make sense! However, when we are pushing or pulling toward the $x$-axis, the function is expanding or contracting vertically - making it a vertical dilation!

$$cf(x)$$

Let's compare a couple of graphs!

Example

First, let's look at the parent function for the quadratic equation. We are going to sketch a graph. Here is the graph of the function of a quadratic.

If we want to stretch the graph, we can multiply the function by a fraction, such as $\frac14$. This is a graph of $f(x)=\frac{1}{4}x^{2}$. The graph of the parent function is in red.

To help understand the concept, you could follow a singular point on the graph and see how that point changes when you transform the original function.

Now, let's add in a function that is compressed from the parent function. Let's add in a graph of $f(x)=4x^2$ in green!

Looking Forward

Our next lesson will cover horizontal compressions and horizontal stretches, also known as horizontal dilations.

If you want to look at this concept the other way, you can also take a graph and write a formula or equation from the graph.