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Chapter 3: Problem 63

Explain the relationship between the degree of a polynomial function and the number of turning points on its graph.

Short Answer

Expert verified
A polynomial function of degree \(n\) has at most \(n-1\) turning points on its graph.

Step by step solution

01

Definition of a Polynomial Function

A polynomial function is a function that can be expressed in the form \(f(x) = a_n x^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_2, a_1, a_0\) are constants, and \(n\) is a nonnegative integer.
02

Degree of a Polynomial

The degree of a polynomial function is the highest power of \(x\) in its expression. It is represented by the \(n\) in the expression given above. For example, in the polynomial \(f(x) = 2x^3 - 7x^2 + 2x + 1\), the degree is 3.
03

Turning Points on a Graph

A turning point on a graph of a function is a point where the graph changes from rising to falling, or falling to rising. These are the local maximums and minimums of the function.
04

Relationship between Degree and Turning Points

The relationship between the degree of a polynomial and the number of turning points on its graph is that a polynomial function of degree \(n\) has at most \(n-1\) turning points. This follows from the fact that each turning point corresponds to a factor of the polynomial, and a polynomial of degree \(n\) can have at most \(n\) factors.

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Most popular questions from this chapter

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=5 x^{4}+7 x^{2}-x+9$$

Use a graphing utility to graph $$f(x)=\frac{x^{2}-4 x+3}{x-2} \quad \text { and} \quad g(x)=\frac{x^{2}-5 x+6}{x-2}$$ What differences do you observe between the graph of \(f\) and \(g ?\) How do you account for these differences?

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-2(x-4)^{2}\left(x^{2}-25\right)$$

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=\frac{x^{2}+7}{3}$$

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-5 x^{4}+7 x^{2}-x+9$$
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