Polynomial Functions

A polynomial function is any function that is a term or a sum (or difference) of terms in which each term is a real number, a variable, or the product of a real number and variable with whole number exponents. In other words, a polynomial function consists of a polynomial.

In this chapter, we will focus on one-variable polynomial functions, like [latex]f(x)=-2x+4[/latex], [latex]g(x)=2x^2-2x+4[/latex], or [latex]h(x)=x^5+2x^3-12x+3[/latex]. Notice that these polynomial functions are written in descending order from the highest exponent to the lowest.

Identifying the Shape of the Graph of a Polynomial Function

Knowing the degree of a polynomial function is useful in helping us predict what its graph will look like. The leading term has the highest exponent on the variable, meaning that the leading term will grow significantly faster than the other terms in the function as [latex]x[/latex] gets very large or very small. So, the behavior of the leading term will dominate the graph.

As an example, we compare the outputs of a degree [latex]2[/latex] polynomial and a degree [latex]5[/latex] polynomial in table 1.

As the inputs for both functions get larger, the degree [latex]5[/latex] polynomial outputs get much larger than the degree [latex]2[/latex] polynomial outputs. This is why we use the leading term to get a rough idea of the behavior of the graphs of polynomial functions as the independent variable gets very large or very small. This is referred to as the end behavior of the graph.

Let’s take a look at the graphs of some polynomial functions and determine any patterns that exist in their graphs.

Figure 1. Degree 1 polynomial function.	Figure 2. Degree 2 polynomial function.
Figure 3. Degree 3 polynomial function.	Figure 4. Degree 4 polynomial function.
Figure 5. Degree 5 polynomial function.	Figure 6. Degree 6 polynomial function

All of the graphs in figures 1 – 6, represent polynomial functions with positive leading coefficients. Notice that figures 1, 3, and 5 show graphs of functions with odd degrees, while figures 2, 4, and 6 show graphs of functions with even degrees. All of the odd degree polynomial functions have graphs that come from the bottom left and end up at the top right. Think John Travolta in Saturday Night Fever–Right arm up, left arm down! On the other hand, all of the even degree polynomial functions have graphs that start at the top left and end up at the top right. Think football touchdown!

If we look at graphs of the parent functions [latex]f(x)=x^n[/latex], we will notice some major similarities.

Odd Degree Polynomials

Figure 7 shows the graphs of [latex]f\left(x\right)={x}^{1},\;f\left(x\right)={x}^{3},\;f\left(x\right)={x}^{5}[/latex], and [latex]f(x)=x^{51}[/latex], which all have odd degrees. When [latex]n=1[/latex] we have the linear function [latex]f(x)=x[/latex] whose graph is a line. The graphs when [latex]n=3, 5, ...[/latex] all have similar end behavior that is right-up and left-down. This is because the value of the function goes to positive infinity as the value of [latex]x[/latex] goes to positive infinity and the value of the function goes to negative infinity as the value of [latex]x[/latex] goes to negative infinity.

Notice that as the power increases, the graphs flatten somewhat near the origin ([latex]-11[/latex]). This is because a proper fraction raised to any odd power has a value between –1 and 1, while an [latex]x[/latex]-value greater than 1 in absolute value will get larger in absolute value as the exponent increases. Notice also that all of the graphs pass through the points (–1, 1), (0, 0), and (1, 1). This is because [latex](-1)^n=-1[/latex] and [latex]1^n=1[/latex] when [latex]n[/latex] is odd, while [latex]0^n=0[/latex] for all positive integer values of [latex]n[/latex].

Figure 7. Odd degree polynomial functions

Even Degree Polynomials

Figure 8 shows the graphs of [latex]f\left(x\right)={x}^{2},\;f\left(x\right)={x}^{4},\;f\left(x\right)={x}^{6}[/latex], and [latex]f(x)=x^{50}[/latex], which all have even degrees. These graphs have similar end behavior that is right-up and left-up. This is because the value of the function goes to positive infinity as the value of [latex]x[/latex] goes to either positive or negative infinity (e.g., a number to an even power, no matter the number is positive or negative, is always positive).

Notice that as the power increases, the graphs flatten somewhat near the origin (between [latex]-11[/latex]). This is because a proper fraction raised to any even power has a value less than one, while an [latex]x[/latex]-value greater than 1 will get larger the higher the value of the exponent. Notice also that the graph of [latex]f(x)=x^n[/latex] always passes through (0, 0), (1, 1) and (–1, 1) when [latex]n[/latex] is even. 

Figure 8. Graph of even degree polynomial functions

Example 1

Identify whether each graph represents a polynomial function that has a degree that is even or odd.

a)

b)

Solution

a) Both arms of this polynomial point upward, therefore the degree must be even.  If we apply negative inputs to an even degree polynomial, you will get positive outputs back.

b) As the inputs of this polynomial become more negative the outputs also become negative. The only way this is possible is with an odd degree polynomial. Therefore, this polynomial must have an odd degree.

Try It 2

Identify whether the leading coefficient is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.

Graph 1

Graph 2

Graph 3

Show Answer

Graph 1: Even degree; negative leading coefficient

Graph 2: Odd degree; negative leading coefficient

Graph 3: Even degree: positive leading coefficient

Zeros

A zero of a function [latex]f(x)[/latex] is any value of the independent variable [latex]x[/latex] that causes the function value to equal zero. i.e., [latex]f(x) = 0[/latex]. On the coordinate plane, zeros are the intersection points between the graph of a function [latex]y=f(x)[/latex] and the [latex]x[/latex]-axis. In other words, they are the [latex]x[/latex]-intercepts. Since [latex]f(x) = 0[/latex], each intersection point has a [latex]y[/latex]-coordinate of 0.

Figure 11. Graph of [latex]h(x)=x^3+4x^2+x-6[/latex]

Figure 11 shows the graph of [latex]h(x)=x^3+4x^2+x-6[/latex]. The graph of the function has [latex]x[/latex]-intercepts at the points (–3, 0), (–2, 0), and (1, 0). Therefore, the zeros of the function are [latex]x = –3, –2[/latex] and [latex]1[/latex].

Notice that when we identify an [latex]x[/latex]-intercept we write it as a point: e.g., (–2, 0). However, when we identify a zero, we write is as a solution of the equation [latex]f(x)=0[/latex]: e.g., [latex]x=–2[/latex].

The Number of Turning Points

When the [latex]y[/latex]-values of a graph increase as [latex]x[/latex] increases, we say the graph is increasing over the given [latex]x[/latex]-values. Likewise, when the [latex]y[/latex]-values of a graph decrease as [latex]x[/latex] increases, we say the graph is decreasing over the given [latex]x[/latex]-values. The point where a graph changes from increasing to decreasing or vice versa is called a turning point. This is illustrated in figure 12 where the graph of the function has 3 turning points.