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Section 9.1 Power Functions

In this activity, we will explore power functions.

A power function is a function of the form

\begin{equation*} f(x) = x^a \end{equation*}

where \(a\) is a constant real number.

Our goal is to learn to predict what the graph of a power function will look like, depending on the exponent \(a\text{.}\)

Positive Integer Exponents.

We will begin by examining power functions where the exponent \(n\) is a positive integer \((1, 2, 3, \cdots )\text{.}\) In order to do this, we should briefly remind ourselves about the difference between even and odd exponents.

Now, let's see the graphs of different power functions with positive exponents, and generalize what happens if the exponent is even or odd.

Use your observations to identify the function in the next exercise.

Now, recall what you know about function transformations from chapter 6 — in particular, that the transformation

\begin{equation*} y = -f(x) \end{equation*}

will reflect the graph of \(f(x)\) over the \(x\)-axis. Use this in the next exercise.

Negative Integer Exponents.

Next, we will expand our set of power functions to include negative integer exponents. First, however, we should review the meaning of a negative exponent.

A positive integer exponent refers to repeated multiplication:

\begin{equation*} x^5 = x\cdot x\cdot x\cdot x\cdot x \end{equation*}

It is often helpful to think of this as a product beginning with the number \(1\text{,}\) so that we really have:

\begin{equation*} x^5 = 1\cdot x\cdot x\cdot x\cdot x\cdot x \end{equation*}

Now, a negative integer exponent refers to repeated division. So, expanding an expression like \(x^{-3}\) is easy if we begin with the number \(1\text{,}\) as in:

\begin{align*} x^{-3} \amp= 1 / (x\cdot x\cdot x)\\ \amp= \frac{1}{x\cdot x\cdot x}\\ \amp= \frac{1}{x^3} \end{align*}

Simplify the following expressions so they have positive exponents:

  • \(x^{-2}\)

  • \(5x^{-3}\)

Solution

  • \(x^{-2} = \frac{1}{x^2}\)

  • \(5x^{-3} = 5\cdot \frac{1}{x^3} = \frac{5}{x^3}\)

So, a variable raised to a negative exponent is the same as dividing by that variable raised to a positive exponent. Keep that in mind as you answer the next exercise.

Now, explore the graphs of power functions with negative integer exponents. You will see a pattern for the shapes of these graphs, depending on whether the exponent is even or odd.

Use your observations to identify the function in the next exercise.

Again, remember the function transformation for reflecting over the \(x\)-axis. Use this in the next exercise.

Fractional Exponents.

We now explore the graphs of power functions which have fractional exponents of the form \(\frac{1}{n}\text{.}\)

Here, we only consider when the exponent is positive.

Before looking at graphs, we will briefly revisit the meaning of a fractional exponent.

Recall the meaning of a fractional exponent. For example, the expression \(9^{\frac{1}{2}}\) is the same as the square root of \(9\text{:}\)

\begin{equation*} 9^{\frac{1}{2}} = \sqrt{9} = 3 \end{equation*}

This can be justified by using properties of exponents. If we multiplied \(9^{\frac{1}{2}}\) by itself, we would have:

\begin{align*} 9^{\frac{1}{2}}\cdot 9^{\frac{1}{2}} \amp= 9^{\frac{1}{2} + \frac{1}{2}}\\ \amp= 9^1\\ \amp= 9 \end{align*}

So, \(9^{\frac{1}{2}}\) must be a square root of \(9\text{,}\) because squaring it actually equals \(9\text{.}\)

Find the following values:

  • \(100^{\frac{1}{2}}\)

  • \(-100^{\frac{1}{2}}\)

  • \((-100)^{\frac{1}{2}}\)

Solution

  • \(100^{\frac{1}{2}} = \sqrt{100} = 10\)

  • \(-100^{\frac{1}{2}} = -\sqrt{100} = -10\)

  • \((-100)^{\frac{1}{2}} = \sqrt{-100}\) is not a real number

Now explore the graphs of power functions with exponents of the form \(\frac{1}{n}\text{,}\) noting what happens when \(n\) is even or odd.

So we see that fractional exponents refer to \(n^{\rm{th}}\) roots: \(x^{\frac{1}{2}}\) is a square root, \(x^{\frac{1}{3}}\) is a cube root, etc. 1 One should be careful when evaluating power functions of this type when the input \(x\) is negative. Some calculators/programs will evaluate an expression like \((-8)^{\frac{1}{3}}\) differently, and actually not return a real number. In this course, we will treat \(x^{\frac{1}{3}}\) and \(\sqrt[3]{x}\) the same.

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