Understand the relationship between degree and turning points

In addition to the end behavior, recall that we can analyze a polynomial function’s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function [latex]f\left(x\right)={x}^{4}-{x}^{3}-4{x}^{2}+4x[/latex] in Figure 11. The graph has three turning points.

Graph of an odd-degree polynomial with a negative leading coefficient. Note that as x goes to positive infinity, f(x) goes to negative infinity, and as x goes to negative infinity, f(x) goes to positive infinity.

Figure 11

This function f is a 4th degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.

A General Note: Interpreting Turning Points

A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).

A polynomial of degree n will have at most n – 1 turning points.

Example 7: Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function

Find the maximum number of turning points of each polynomial function.

  1. [latex]f\left(x\right)=-{x}^{3}+4{x}^{5}-3{x}^{2}++1[/latex]
  2. [latex]f\left(x\right)=-{\left(x - 1\right)}^{2}\left(1+2{x}^{2}\right)[/latex]

Solution

  1. [latex]f\left(x\right)=-x{}^{3}+4{x}^{5}-3{x}^{2}++1[/latex]

    First, rewrite the polynomial function in descending order: [latex]f\left(x\right)=4{x}^{5}-{x}^{3}-3{x}^{2}++1[/latex]

    Identify the degree of the polynomial function. This polynomial function is of degree 5.

    The maximum number of turning points is 5 – 1 = 4.

  2. [latex]f\left(x\right)=-{\left(x - 1\right)}^{2}\left(1+2{x}^{2}\right)[/latex]
Graph of f(x)=x^4-x^3-4x^2+4x which denotes where the function increases and decreases and its turning points.

Figure 12

First, identify the leading term of the polynomial function if the function were expanded.

Then, identify the degree of the polynomial function. This polynomial function is of degree 4.

The maximum number of turning points is 4 – 1 = 3.