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## Section9.1Power 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.

###### Exercise9.1.1
WeBWorK Exercise

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
WeBWorK Exercise

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

###### Exercise9.1.3
WeBWorK 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.

WeBWorK 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.

###### Example9.1.10

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.

###### Exercise9.1.11
WeBWorK 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.

###### Exercise9.1.12
WeBWorK Exercise

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

###### Exercise9.1.13
WeBWorK Exercise

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

WeBWorK 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.

###### Example9.1.20

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.

###### Exercise9.1.21
WeBWorK Exercise

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.