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## SectionHorizontal Stretches and Compressions

In the previous section we discussed the result of multiplying the output of the function by a constant value. However, what happens when we multiply the input of the function? To explore this idea, we look at the graphs of

\begin{equation*} f(x)=(x-1)^2 \text{, } ~f(2x)=(2x-1)^2\ \text{ ,and } ~f\left(\dfrac{1}{2}x\right)=\left(\dfrac{1}{2}x-1\right)^2 \end{equation*}

and discuss how they are related. $x$ $y=f(x)$ $y=f(2x)$ $-1$ $4$ $8$ $-.5$ $2.25$ $4$ $0$ $1$ $1$ $.5$ $.25$ $0$ $1$ $0$ $1$ $2$ $1$ $9$

As we can see above, compared to the graph of $f(x)\text{,}$ the graph of $f(2x)$ is compressed horizontally by a factor of $2 \text{.}$ Effectively, if we are given a point $(x,y)$ on the graph of $f(x)$ then $\left(\dfrac{1}{2}x,y\right)$ is a point on the graph of $f(2x)\text{.}$

Looking at the table above we can verify this for a few points. For example, the point $(2,1)$ is on the graph of $f(x)\text{.}$ Then

\begin{equation*} \left(\dfrac{1}{2}(2),1\right)=(1,1) \end{equation*}

is a point on the graph $f(2x)\text{.}$

###### Horizontal Stretch $x$ $y=f(x)$ $y=f\left(\dfrac{1}{2}x\right)$ $-1$ $4$ $2.25$ $0$ $1$ $1$ $1$ $0$ $.25$ $2$ $1$ $0$ $4$ $9$ $1$

The graph of $f\left(\dfrac{1}{2}x\right)$ is stretched horizontally by a factor of $2$ compared to the graph of $f(x) \text{.}$ Further, if $(x,y)$ is a point on the graph of $f(x)\text{,}$ then $(2x,y)$ is a point on the graph of $f\left(\dfrac{1}{2}x\right)\text{.}$

We can see this playing out in our example above. Notice that $(2,1)$ is a point on $f(x)\text{,}$ and

\begin{equation*} (2(2),1)=(4,1) \end{equation*}

is a point on the graph of $f\left(\dfrac{1}{2}x\right)$ as shown in the table and graph above. In general we have:

###### Horizontal Stretches, Compressions, and Reflections

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

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

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

###### Example272 The graph of $f(x)$ is shown along with either a horizontal stretch of compression of $f(x) \text{.}$ Decide if $g(x)$ is a stretch or a compression, and give a formula for $g(x)$ in terms of $f(x)\text{.}$

Solution

First, notice that the $y$-intercept stays fixed while the $x$-intercepts shift closer to the $y$-axis. This tells us that $g(x)$ is a horizontal compression. The $x$-intercepts of $f(x)$ are $x=-1,1,2$ while the $x$-intercepts of $g(x)$ are $x=-.5,.5,1\text{.}$

So, the $x$-intercepts of $g(x)$ can be achieved by taking the intercepts of $f(x)$ and divide each by 2. This tells us that $g(x)$ is a horizontal compression by a factor of $2 \text{.}$ Hence, we may write

\begin{equation*} g(x)=f(2x). \end{equation*}