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 \(a\text{.}\)

We will begin by examining power functions with integer exponents.

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

Now, recall what you know about function transformations from chapter 8 — 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.

Next, we will expand our set of power functions to include negative integer exponents.

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.

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

In this case we will look first at the case where the exponent is positive.