## Section10.2Transformations

### Subsection10.2.1Review Material

In this activity, you will review some basics from previous courses:

• Vertical and Horizontal Translations

• Vertical Stretches and Compressions

In the first exercise below, you will encounter your first of many WeBWorK problems. Click to open the WeBWorK exercise, and you will see a graph that you can manipulate. Answer the questions that follow, and click to submit your answers.

Next, recall how a graph changes when we either add or subtract from the input of the function.

Finally, explore the effect of multiplying the output of a function by a constant number.

Questions you should be able to answer:

1. In general, how does the graph change when you add a number to the output of a function?

2. In general, how does the graph change when you subtract a number from the output of a function?

Practice using these ideas in the next two exercises.

Finally, we'll review an application of using vertical and horizontal translations to quickly perform a familiar task: Finding the equation of a line

### Subsection10.2.2Horizontal Stretches/Compressions

We have already seen that the graph of a function may be stretched or compressed vertically by multiplying the output of the function by a constant number $a\text{.}$

• If $a>1\text{,}$ then graphing $y = af(x)$ would be the result of stretching $y = f(x)$ vertically, away from the $x$–axis. All of the $y$ values on the graph of $f(x)$ would be multiplied by this number $a\text{,}$ so they would now be larger in magnitude (either positive or negative). Thus, the points would be stretched farther away from the $x$–axis.

• If $0<a<1\text{,}$ then the graph of $y = af(x)$ would be like compressing the graph of $y = f(x)$ vertically, toward the $x$–axis. All of the $y$ values on the graph of $f(x)$ would be multiplied by the number $a\text{,}$ so they would become smaller in magnitude. Thus, the points would be compressed toward the $x$–axis.

Finally, we know that adding or subtracting a number from the input of a function will shift the graph either left or right.

##### Horizontal Stretches and Compressions.

Now, we will complete the story by investigating how to stretch or compress a function horizontally.

It may have surprised you to see that multiplying the input of a function by a number larger than $1$ actually made the graph compress horizontally.

Why did this happen? Let's “crunch some numbers” for a function and one of its transformations.

The table below shows some values for the function $f(x)$ and its transformation $f(2x)\text{.}$

 $x$ $f(x)$ $f(2x)$ $1$ $8$ $2$ $13$ $23$ $3$ $18$ $4$ $23$ $5$ $28$ $6$ $33$

Complete the values for the transformation $f(2x)\text{.}$

For any value of the input $x\text{,}$ evaluating $f(2x)$ means you're looking at the output of $f$ farther down the table, where the input is $2x\text{.}$

So, when $x = 2\text{,}$ the output of $f(2x)$ is the same as $f(4) = 23\text{.}$

The completed table looks like:

 $x$ $f(x)$ $f(2x)$ $1$ $8$ 13 $2$ $13$ $23$ $3$ $18$ 33 $4$ $23$ 43 $5$ $28$ 53 $6$ $33$ 63

[The function $f$ is linear, and its pattern can be used to determine values of the table.]

Notice that the output values for $f(x)\text{,}$ which occur later in the table, now happen earlier for the transformation $f(2x)\text{.}$ This makes sense graphically, as the points at certain heights now occur earlier, at smaller $x$–values for $f(2x)\text{.}$

###### Note10.2.12.

It is useful to realize that a transformation like $f(0.25x)$ may be written as:

\begin{equation*} f\left(\frac{x}{4}\right) \end{equation*}

Any transformation of the type $f(ax)$ may be rewritten as $f\left(\dfrac{x}{\sfrac{1}{a}}\right)\text{,}$ where one of the numbers, $a$ or $\frac{1}{a}\text{,}$ is greater than $1\text{.}$

Therefore, another way to remember what this type of transformation does is the following.

If you rewrite the transformation so that $a>1\text{,}$ then:

$f(ax)$

Compresses the graph of $f$ by a factor of $a$

“Multiplying the input variable by a number greater than $1$ will compress the graph.”

Example: $f(3x)$ is only $1/3$ as wide as $f(x)\text{.}$

$f\big(\frac{x}{a}\big)$

Stretches the graph of $f$ by a factor of $a$

Example? $f\left(\frac{x}{4}\right)$ is $4$ times as wide as $f(x)\text{.}$

Often, we will want to horizontally stretch or compress a function by a certain factor. Or, if we already have a transformed function's graph, we may need to determine what the stretching/compressing factor was.

The graph at left shows $y = f(x)\text{.}$ This graph was horizontally stretched to make the graph $y = f(ax)$ at right.

The objective is to find what value of $a$ caused the horizontal stretch to the graph of $f\text{.}$

1. Since this transformation stretches points away from the $x$–axis, fix your attention on the point $(2, 0)$ on the graph of $f\text{.}$

Its corresponding point $(5, 0)$ on the graph of $g$ has been stretched by what factor away from the $x$–axis? 1 That is, how many times larger is the $x$–value of the stretched point?

2. Using the idea from Note 10.2.12, determine the value of $a$ if the stretched graph is $f\left(\frac{x}{a}\right)\text{.}$

Solution

The points on $g(x)$ are $\frac{5}{2}=2.5$ times as far from the $x$–axis as those on the graph of $f(x)\text{.}$

Since the graph was stretched, then we know that the transformation is $f\left(\frac{x}{a}\right)$ where $a = 2.5\text{.}$ So, the transformation could be written as either:

\begin{equation*} y = f\left(\frac{x}{2.5}\right) \end{equation*}

Or

\begin{equation*} y = f\left(0.4x\right) \end{equation*}
##### Stretching or Compressing a standard function.

You are already familiar with many basic functions:

 $y = x$ $y = x^2$ $y = e^x$ $y = log_{10}(x)$ $y = \frac{1}{x}$

If you know these basic shapes, you can create infinite variations by using appropriate transformations.

In the next exercise, you will be given a function and its formula. Then use what you know about transformations in order to choose the correct formula for the transformed graph.