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.

#####
Exercise9.1.1

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

#####
Exercise9.1.2

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

#####
Exercise9.1.3

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.

#####
Exercise9.1.4

##### 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.

#####
Example9.1.5

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:

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.

#####
Exercise9.1.6

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*.

#####
Exercise9.1.7

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

#####
Exercise9.1.8

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

#####
Exercise9.1.9

##### 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.

#####
Example9.1.10

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

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

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

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

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.

#####
Exercise9.1.11

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.