
## SectionVertical Stretches and Compressions

We have seen that adding a constant to the expression defining a function results in a translation of its graph. What happens if we multiply the expression by a constant? Consider the graphs of the functions

\begin{equation*} f(x)= 2x^2 \text{, and } ~g(x)= \frac{1}{2}x^2 \end{equation*}

shown in Figure259, and Figure260. We will compare each to the graph of $y = x^2\text{.}$

 $x$ $y=x^2$ $f(x)=2x^2$ $-2$ $4$ $8$ $-1$ $1$ $2$ $0$ $0$ $0$ $1$ $1$ $2$ $2$ $4$ $8$

Compared to the graph of $y = x^2\text{,}$ the graph of $f (x) = 2x^2$ is expanded, or stretched, vertically by a factor of $2\text{.}$ The $y$-coordinate of each point on the graph has been doubled, as you can see in the table of values, so each point on the graph of $f$ is twice as far from the $x$-axis as its counterpart on the basic graph $y = x^2\text{.}$

 $x$ $y=x^2$ $g(x)=\frac{1}{2}x^2$ $-2$ $4$ $2$ $-1$ $1$ $\frac{1}{2}$ $0$ $0$ $0$ $1$ $1$ $\frac{1}{2}$ $2$ $4$ $2$

The graph of $g(x) = \dfrac{1}{2}x^2$ is compressed vertically by a factor of $2\text{;}$ each point is half as far from the $x$-axis as its counterpart on the graph of $y = x^2\text{.}$

In general, we have the following principles.

###### Vertical Stretches, Compressions, and Reflections

Compared with the graph of $y = f (x)\text{,}$ the graph of $y = a f (x)\text{,}$ where $a \ne 0\text{,}$ is

1. stretched vertically by a factor of $\abs{a}$ if $\abs{a}\gt 1\text{,}$
2. compressed vertically by a factor of $\frac{1}{\abs{a}}$ if $0\lt\abs{a}\lt 1\text{,}$ and
3. reflected about the $x$-axis (and stretched or compressed) if $a\lt 0\text{.}$

As you may have notice by now through our examples, a vertical stretch or compression will never change the $x$ intercepts. This is a good way to tell if such a transformation has occurred.

###### Example261

Graph the following functions.

1. $g(x) = 3\sqrt[3]{x}$
2. $h(x) =\dfrac{-1}{2}\abs{x}$
Solution
1. The graph of $g(x) = 3\sqrt[3]{x}$ is a vertical stretch of the basic graph $y = \sqrt[3]{x}$ by a factor of $3\text{,}$ as shown in Figure262. Each point on the basic graph has its $y$-coordinate tripled.

2. The graph of $h(x) = \dfrac{-1}{2}\abs{x}$ is a vertical compression of the basic graph $y = \abs{x}$ by a factor of $2\text{,}$ combined with a reflection about the $x$-axis. You may find it helpful to graph the function in two steps, as shown in Figure263.

###### Example264

The function $A = f (t)$ graphed in Figure265 gives a person's blood alcohol level $t$ hours after drinking a martini. Sketch a graph of $g(t) = 2 f (t)$ and explain what it tells you.

Solution

To sketch a graph of $g\text{,}$ we stretch the graph of $f$ vertically by a factor of $2\text{,}$ as shown in Figure266. At each time $t\text{,}$ the person's blood alcohol level is twice the value given by $f\text{.}$ The function $g$ could represent a person's blood alcohol level $t$ hours after drinking two martinis.

###### Example267

If the Earth were not tilted on its axis, there would be 12 daylight hours every day all over the planet. But in fact, the length of a day in a particular location depends on the latitude and the time of year.

The graph in Figure268 shows $H = f (t)\text{,}$ the length of a day in Helsinki, Finland, $t$ days after January 1, and $R = g(t)\text{,}$ the length of a day in Rome. Each is expressed as the number of hours greater or less than 12. Write a formula for $f$ in terms of $g\text{.}$ What does this formula tell you?

Solution

$f (t)\approx 2g(t)\text{.}$ On any given day, the number of daylight hours varies from $12$ hours about twice as much in Helsinki as it does in Rome.

In the following applet, explore the properties of vertical stretches and compressions.