Precalculus: An Active Reading Approach: Elementary Functions

Subsection 10.2.1 Review 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

Subsection 10.2.2 Horizontal 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.

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.

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 \(f(2x)\text{.}\)

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?¹That 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*}