Identifying Local Behavior of Polynomial Functions

In addition to the end behavior of polynomial functions, we are also interested in what happens in the “middle” of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

We are also interested in the intercepts. As with all functions, the y-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one y-intercept [latex]\left(0,{a}_{0}\right)[/latex]. The x-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one x-intercept.

Determining the Number of Turning Points and Intercepts from the Degree of the Polynomial

A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

The degree of a polynomial function helps us to determine the number of x-intercepts and the number of turning points. A polynomial function of nth degree is the product of n factors, so it will have at most n roots or zeros, or x-intercepts. The graph of the polynomial function of degree n must have at most n – 1 turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.

The following video gives a 5 minute lesson on how to determine the number of intercepts and turning points of a polynomial function given it’s degree.

Using the Principle of Zero Products to Find the Roots of a Polynomial in Factored Form

The Principle of Zero Products states that if the product of n numbers is 0, then at least one of the factors is 0. If [latex]ab=0[/latex], then either [latex]a=0[/latex] or [latex]b=0[/latex], or both a and b are 0. We will use this idea to find the zeros of a polynomial that is either in factored form, or can be written in factored form. For example, the polynomial

[latex]P(x)=(x-4)^2(x+1)(x-7)[/latex]

is in factored form. In the following examples, we will show the process of factoring a polynomial, and calculating it’s x and y-intercepts.

Example: Determining the Intercepts of a Polynomial Function

Given the polynomial function [latex]f\left(x\right)=\left(x - 2\right)\left(x+1\right)\left(x - 4\right)[/latex], written in factored form for your convenience, determine the y– and x-intercepts.

Solution

The y-intercept occurs when the input is zero so substitute 0 for x.

[latex]\begin{array}{c}f\left(0\right)=\left(0 - 2\right)\left(0+1\right)\left(0 - 4\right)\hfill \\ \text{ }=\left(-2\right)\left(1\right)\left(-4\right)\hfill \\ \text{ }=8\hfill \end{array}[/latex]

The y-intercept is (0, 8).

The x-intercepts occur when the output is zero.

[latex]\begin{array}{c}\text{ }0=\left(x - 2\right)\left(x+1\right)\left(x - 4\right)\hfill \\ x - 2=0\hfill & \hfill & \text{or}\hfill & \hfill & x+1=0\hfill & \hfill & \text{or}\hfill & \hfill & x - 4=0\hfill \\ \text{ }x=2\hfill & \hfill & \text{or}\hfill & \hfill & \text{ }x=-1\hfill & \hfill & \text{or}\hfill & \hfill & x=4 \end{array}[/latex]

The x-intercepts are [latex]\left(2,0\right),\left(-1,0\right)[/latex], and [latex]\left(4,0\right)[/latex].

We can see these intercepts on the graph of the function shown below.

Example: Determining the Intercepts of a Polynomial Function with Factoring

Given the polynomial function [latex]f\left(x\right)={x}^{4}-4{x}^{2}-45[/latex], determine the y– and x-intercepts.

Solution

The y-intercept occurs when the input is zero.

[latex]\begin{array}{c} \\ f\left(0\right)={\left(0\right)}^{4}-4{\left(0\right)}^{2}-45\hfill \hfill \\ \text{ }=-45\hfill \end{array}[/latex]

The y-intercept is [latex]\left(0,-45\right)[/latex].

The x-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.

[latex]\begin{array}{c}f\left(x\right)={x}^{4}-4{x}^{2}-45\hfill \\ =\left({x}^{2}-9\right)\left({x}^{2}+5\right)\hfill \\ =\left(x - 3\right)\left(x+3\right)\left({x}^{2}+5\right)\hfill \end{array}[/latex]

[latex]0=\left(x - 3\right)\left(x+3\right)\left({x}^{2}+5\right)[/latex]

[latex]\begin{array}{c}x - 3=0\hfill & \text{or}\hfill & x+3=0\hfill & \text{or}\hfill & {x}^{2}+5=0\hfill \\ \text{ }x=3\hfill & \text{or}\hfill & \text{ }x=-3\hfill & \text{or}\hfill & \text{(no real solution)}\hfill \end{array}[/latex]

The x-intercepts are [latex]\left(3,0\right)[/latex] and [latex]\left(-3,0\right)[/latex].

We can see these intercepts on the graph of the function shown below. We can see that the function is even because [latex]f\left(x\right)=f\left(-x\right)[/latex].

The Whole Picture

Now we can bring the two concepts of turning points and intercepts together to get a general picture of the behavior of polynomial functions.  These types of analyses on polynomials developed before the advent of mass computing as a way to quickly understand the general behavior of a polynomial function. We now have a quick way, with computers, to graph and calculate important characteristics of polynomials that once took a lot of algebra.

In the first example, we will determine the least degree of a polynomial based on the number of turning points and intercepts.

Example: Drawing Conclusions about a Polynomial Function from the Graph

Given the graph of the polynomial function below, determine the least possible degree of the polynomial, and whether it is even or odd. Use end behavior, the number of intercepts, and the number of turning points to help you.

Solution

The end behavior of the graph tells us this is the graph of an even-degree polynomial.

The graph has 2 x-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.

Now you try to determine the least possible degree of a polynomial given it’s graph.

Try It

Given the graph of the polynomial function below, determine the least possible degree of the polynomial, and whether it is even or odd. Use end behavior, the number of intercepts, and the number of turning points to help you.

Solution

The end behavior indicates an odd-degree polynomial function; there are 3 x-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.

Now we will show that you can also determine the least possible degree and number of turning points of a polynomial function given in factored form.

Now it’s your turn to determine the local behavior of a polynomial given in factored form.