3.3 Power functions and polynomial functions

Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in [link] .

The population can be estimated using the function $P(t)=-0.3t^3+97t+800,$ where $P(t)$ represents the bird population on the island $t$ 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

In order to better understand the bird problem, we need to understand a specific 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. (A number that multiplies a variable raised to an exponent is known as a coefficient.)

As an example, consider functions for area or volume. The function for the area of a circle with radius $r$ is

and the function for the volume of a sphere with radius $r$ is

Both of these are examples of power functions because they consist of a coefficient, $\pi$ or $\frac{4}{3}\pi ,$ multiplied by a variable $r$ raised to a power.

Identifying end behavior of power functions

[link] shows the graphs of $f(x)=x^2,g(x)=x^4$ and $\text{and}h(x)=x^6,$ which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.