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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 Figure257, and Figure258. We will compare each to the graph of \(y = x^2\text{.}\)

2 x-sq and basic parabola
\(x\) \(y=x^2\) \(f(x)=2x^2\)
\(-2\) \(4\) \(8\)
\(-1\) \(1\) \(2\)
\(0\) \(0\) \(0\)
\(1\) \(1\) \(2\)
\(2\) \(4\) \(8\)
Figure257

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 \(y = x^2\text{.}\)

half x-sq and basic parabola
\(x\) \(y=x^2\) \(g(x)=\frac{1}{2}x^2\)
\(-2\) \(4\) \(2\)
\(-1\) \(1\) \(\frac{1}{2}\)
\(0\) \(0\) \(0\)
\(1\) \(1\) \(\frac{1}{2}\)
\(2\) \(4\) \(2\)
Figure258

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.

Example259

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 Figure260. Each point on the basic graph has its \(y\)-coordinate tripled.

    scale cube root
    Figure260
  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 Figure261.

    scale absolute value
    Figure261
  1. Graph the function \(f (x) = 2\abs{x}\text{.}\)
  2. How is the graph of \(f\) different from the graph of \(y =\abs{x}\text{?}\)
Example263

The function \(A = f (t)\) graphed in Figure264 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.

alcohol level after alcohol
Figure264
Solution

To sketch a graph of \(g\text{,}\) we stretch the graph of \(f\) vertically by a factor of \(2\text{,}\) as shown in Figure265. 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.

alcohol level after more alcohol
Figure265

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 Figure267 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?

daylight hours at two latitudes
Figure267
Figure268Explore the properties of vertical stretches and compressions discussed in this section with this applet. You can change the base function \(f(x)\) using the input box and see many different stretches/compressions of \(f(x)\) by moving around the \(a\) slider.