30 Power Functions and Polynomial Functions
Learning Objectives
In this section, you will:
- Identify power functions.
- Identify end behavior of power functions.
- Identify polynomial functions.
- Identify the degree and leading coefficient of polynomial functions.
Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in (Figure).
| Year | [latex]2009[/latex] | [latex]2010[/latex] | [latex]2011[/latex] | [latex]2012[/latex] | [latex]2013[/latex] |
| Bird Population | [latex]800[/latex] | [latex]897[/latex] | [latex]992[/latex] | [latex]1,083[/latex] | [latex]1,169[/latex] |
The population can be estimated using the function[latex]\,P\left(t\right)=-0.3{t}^{3}+97t+800,\,[/latex]where[latex]\,P\left(t\right)\,[/latex]represents the bird population on the island[latex]\,t\,[/latex]years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.
Identifying Power Functions
Before we can understand the bird problem, it will be helpful to understand a different type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number.
As an example, consider functions for area or volume. The function for the area of a circle with radius[latex]\,r\,[/latex]
is
and the function for the volume of a sphere with radius[latex]\,r\,[/latex]
is
Both of these are examples of power functions because they consist of a coefficient,[latex]\,\pi \,[/latex]or[latex]\,\frac{4}{3}\pi ,\,[/latex]multiplied by a variable[latex]\,r\,[/latex]raised to a power.
Power Function
A power function is a function that can be represented in the form
where[latex]\,k\,[/latex]
and[latex]\,p\,[/latex]are real numbers, and[latex]\,k\,[/latex]
is known as the coefficient.
Is[latex]\,f\left(x\right)={2}^{x}\,[/latex]a power function?
No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.
Identifying Power Functions
Which of the following functions are power functions?
[latex]\begin{array}{cccc}\hfill f\left(x\right)& =& 1\hfill & \phantom{\rule{2em}{0ex}}\text{Constant function}\hfill \\ \hfill f\left(x\right)& =& x\hfill & \phantom{\rule{2em}{0ex}}\text{Identify function}\hfill \\ \hfill f\left(x\right)& =& {x}^{2}\hfill & \phantom{\rule{2em}{0ex}}\text{Quadratic function}\hfill \\ \hfill f\left(x\right)& =& {x}^{3}\hfill & \phantom{\rule{2em}{0ex}}\text{Cubic function}\hfill \\ \hfill f\left(x\right)& =& \frac{1}{x}\hfill & \phantom{\rule{2em}{0ex}}\text{Reciprocal function}\hfill \\ \hfill f\left(x\right)& =& \frac{1}{{x}^{2}}\hfill & \phantom{\rule{2em}{0ex}}\text{