Identify polynomial functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius r of the spill depends on the number of weeks w that have passed. This relationship is linear.

[latex]r\left(w\right)=24+8w[/latex]

We can combine this with the formula for the area A of a circle.

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

Composing these functions gives a formula for the area in terms of weeks.

[latex]\begin{array}{c}A\left(w\right)=A\left(r\left(w\right)\right)\\ =A\left(24+8w\right)\\ =\pi {\left(24+8w\right)}^{2}\end{array}[/latex]

Multiplying gives the formula.

[latex]A\left(w\right)=576\pi +384\pi w+64\pi {w}^{2}[/latex]

This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

Define the degree and leading coefficient of a polynomial function

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.

A General Note: Terminology of Polynomial Functions

We often rearrange polynomials so that the powers are descending.

When a polynomial is written in this way, we say that it is in general form.

In the following video, we show more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.

Identifying End Behavior of Polynomial Functions

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree.

Polynomial Function	Leading Term	Graph of Polynomial Function
[latex]f\left(x\right)=5{x}^{4}+2{x}^{3}-x - 4[/latex]	[latex]5{x}^{4}[/latex]
[latex]f\left(x\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[/latex]	[latex]-2{x}^{6}[/latex]
[latex]f\left(x\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[/latex]	[latex]3{x}^{5}[/latex]
[latex]f\left(x\right)=-6{x}^{3}+7{x}^{2}+3x+1[/latex]	[latex]-6{x}^{3}[/latex]

Example: Identifying End Behavior and Degree of a Polynomial Function

Describe the end behavior and determine a possible degree of the polynomial function in the graph below.

Solution

As the input values x get very large, the output values [latex]f\left(x\right)[/latex] increase without bound. As the input values x get very small, the output values [latex]f\left(x\right)[/latex] decrease without bound. We can describe the end behavior symbolically by writing

[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]

In words, we could say that as x values approach infinity, the function values approach infinity, and as x values approach negative infinity, the function values approach negative infinity.

We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.

In the following video, we show more examples that summarize the end behavior of a polynomial functions, and which components of the function contribute to it.

Try It

Describe the end behavior of the polynomial function in the graph below.

Solution

As [latex]x\to \infty , f\left(x\right)\to -\infty ; as x\to -\infty , f\left(x\right)\to -\infty [/latex]. It has the shape of an even degree power function with a negative coefficient.