7.2: Power Functions
Objectives:
- Determine whether a given function is a power function by rewriting it in the form $f(x) = kx^p$
- Classify power functions and their graphs into four basic types
- Describe the end behavior of power functions
Important Items
Definitions:
power function
Notes:
This lesson should prepare students to learn about polynomials and long-run behavior of rational functions. We will not work with power functions with fractional exponents $p$ or discuss direct or inverse proportionality.
Resources
Here is a review sheet for the four classifications of graphs of power functions put together by a 101 instructor. You can find the tex file for this in Box in the Resources folder.
Lesson Guide
Warm-Up
Have students do Problem 1.
Determine whether a given function is a power function by rewriting it in the form $f(x) = kx^p$
A power function is a function that can be written in the form $f(x) = kx^p$, where $k$ and $p$ are any constants.
Give several examples and non-examples of power functions. Give an example of a power function that is not yet in the form $y=kx^p,$ and demonstrate how to manipulate the function into this form to prove that it is indeed a power function.
- Example:
The area of a circle is a function of the radius: $A(r) = \pi r^2$. To see that this is a power function, note that $k=\pi$ and $p=2$.
- Example:
- Example:
Point out that $p$ can be negative, $k$ can involve a number like $e$ or be a fraction, and $x$ cannot be in an exponent.
Have students do Problems 2 and 3.
Classify power functions and their graphs into four basic types
Summarize the four types of power functions as follows. For now, consider functions with $k=1$ to focus on the effect of $p$, where $p$ is an integer.
\begin{tabular}{|c|c|c|}
\hline
& $p$ Even & $p$ Odd \hline
$p>0$ & &
& \includegraphics{images/section11_1graph1.png} & \includegraphics{images/section11_1graph2.png}
& e.g. $y = x^2$, $y= x^4$ & e.g. $y = x^3$, $y=x^5$ [1ex] \hline
$p<0$ & &
& \includegraphics{images/section11_1graph3.png} & \includegraphics{images/section11_1graph4.png}
& e.g. $y = x^{-2} = \frac{1}{x^2}$, $y=\frac{1}{x^4}$ & e.g. $y = x^{-1} = \frac{1}{x}$, $y = \frac{1}{x^3}$ [1ex] \hline
\end{tabular}
To extend the above examples of even and odd functions to power functions with leading coefficient different than $k=1,$ remind students of how the constant $k$ affects a power function using what they know from Chapter 5.
Have students do Problem 4 and 5.
Describe the end behavior of power functions
Do several examples to introduce end behavior using the notation in Problem 6.
- Example:
- Example:
Have students do Problem 6 and 7. Note that Problem 7 might seem challenging to them at first.
