(credit: Jason Bay, Flickr)

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

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 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.

Identify 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

[latex]A\left(r\right)=\pi {r}^{2}[/latex]

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

[latex]V\left(r\right)=\frac{4}{3}\pi {r}^{3}[/latex]

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 r raised to a power.

Identify end behavior of power functions

The graph below shows the graphs of [latex]f\left(x\right)={x}^{2},g\left(x\right)={x}^{4}[/latex], [latex]h\left(x\right)={x}^{6}[/latex], [latex]k(x)=x^8[/latex], and [latex]p(x)=x^10[/latex] 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.

https://www.desmos.com/calculator/rgdspbzldy

To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol [latex]\infty[/latex] for positive infinity and [latex]-\infty[/latex] for negative infinity. When we say that “x approaches infinity,” which can be symbolically written as [latex]x\to \infty[/latex], we are describing a behavior; we are saying that x is increasing without bound.

With the even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as [latex]x[/latex] approaches positive or negative infinity, the [latex]f\left(x\right)[/latex] values increase without bound. In symbolic form, we could write

[latex]\text{as }x\to \pm \infty , f\left(x\right)\to \infty[/latex]

The graph below shows [latex]f\left(x\right)={x}^{3},g\left(x\right)={x}^{5},h\left(x\right)={x}^{7},k\left(x\right)={x}^{9},\text{and }p\left(x\right)={x}^{11}[/latex], which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.

https://www.desmos.com/calculator/4aeczmnp1w

These examples illustrate that functions of the form [latex]f\left(x\right)={x}^{n}[/latex] reveal symmetry of one kind or another. First, in the even-powered function we see that even functions of the form [latex]f\left(x\right)={x}^{n}\text{, }n\text{ even,}[/latex] are symmetric about the y-axis. In the odd-power function we see that odd functions of the form [latex]f\left(x\right)={x}^{n}\text{, }n\text{ odd,}[/latex] are symmetric about the origin.

For these odd power functions, as x approaches negative infinity, [latex]f\left(x\right)[/latex] decreases without bound. As x approaches positive infinity, [latex]f\left(x\right)[/latex] increases without bound. In symbolic form we write

[latex]\begin{array}{c}\text{as } x\to -\infty , f\left(x\right)\to -\infty \\ \text{as } x\to \infty , f\left(x\right)\to \infty \end{array}[/latex]

The behavior of the graph of a function as the input values get very small ( [latex]x\to -\infty[/latex] ) and get very large ( [latex]x\to \infty[/latex] ) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.

The table below shows the end behavior of power functions in the form [latex]f\left(x\right)=k{x}^{n}[/latex] where [latex]n[/latex] is a non-negative integer depending on the power and the constant.

	Even power	Odd power
Positive constant k > 0
Negative constant k < 0

Example: Identifying the End Behavior of a Power Function

Describe the end behavior of the graph of [latex]f\left(x\right)={x}^{8}[/latex].

Solution

The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As x approaches infinity, the output (value of [latex]f\left(x\right)[/latex] ) increases without bound. We write as [latex]x\to \infty , f\left(x\right)\to \infty [/latex]. As x approaches negative infinity, the output increases without bound. In symbolic form, as [latex]x\to -\infty , f\left(x\right)\to \infty[/latex]. We can graphically represent the function.

Example: Identifying the End Behavior of a Power Function.

Describe the end behavior of the graph of [latex]f\left(x\right)=-{x}^{9}[/latex].

Solution

The exponent of the power function is 9 (an odd number). Because the coefficient is –1 (negative), the graph is the reflection about the x-axis of the graph of [latex]f\left(x\right)={x}^{9}[/latex]. The graph shows that as x approaches infinity, the output decreases without bound. As x approaches negative infinity, the output increases without bound. In symbolic form, we would write

[latex]\begin{array}{c}\text{as } x\to -\infty , f\left(x\right)\to \infty \\ \text{as } x\to \infty , f\left(x\right)\to -\infty \end{array}[/latex]

Analysis of the Solution

We can check our work by using the table feature on Desmos.

1. Enter the function [latex]f\left(x\right)=-{x}^{9}[/latex] into Desmos
2. Create a table with the following x values, and observe the sign of the outputs. [latex]-10,-5,0,5,10[/latex]
3. Now, enter the function [latex]g\left(x\right)={x}^{9}[/latex], and create a similar table. Compare the signs of the outputs for both functions.