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.

Figure 11

This function f is a 4^th 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.

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]

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.

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