## Section 9.1 Power Functions

¶In this activity, we will explore power functions.

A power function is a function of the form

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.

###### Checkpoint 9.1.1.

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

###### Checkpoint 9.1.2.

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

###### Checkpoint 9.1.3.

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

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

###### Checkpoint 9.1.4.

##### 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.

###### Example 9.1.5.

A positive integer exponent refers to *repeated multiplication*:

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

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:

Simplify the following expressions so they have positive exponents:

\(x^{-2}\)

\(5x^{-3}\)

\(x^{-2} = \frac{1}{x^2}\)

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

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.

###### Checkpoint 9.1.6.

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*.

###### Checkpoint 9.1.7.

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

###### Checkpoint 9.1.8.

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

###### Checkpoint 9.1.9.

##### 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.

###### Example 9.1.10.

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

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

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

Find the following values:

\(100^{\frac{1}{2}}\)

\(-100^{\frac{1}{2}}\)

\((-100)^{\frac{1}{2}}\)

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

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

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

###### Checkpoint 9.1.11.

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.