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[[7.1: Combining Functions | Prior Lesson]] | [[8.1: Polynomial Functions | Next Lesson]]
==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==
[[Media:Section 11-2 ClassificationTable.pdf|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.

7.2: Power Functions - Revision history

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