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Chapter 3: Problem 85

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

Short Answer

Expert verified
The degree of a polynomial function influences the maximum number of turning points in its graph. A polynomial of degree \(n\) can have up to \(n-1\) turning points.

Step by step solution

01

Introduction

A polynomial function is an expression of the form \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0\), where each \(a_i\) is a constant, \(n\) is a nonnegative integer, and \(a_n\) is not zero. The highest power in the polynomial is called the degree of the polynomial.
02

Turning points

Turning points refer to the points on a graph of a function where the graph changes direction, i.e., it shifts from increasing to decreasing or decreasing to increasing.
03

Relationship

The degree of a polynomial function majorly influences the number of turning points on its graph. For a polynomial of degree \(n\), the maximum number of turning points the graph will have is \(n-1\). This is, however, an upper limit, the graph could have fewer turning points as well.
04

Example

For instance, if the degree of the polynomial function is 3 (i.e., a cubic function), then the maximum number of turning points the function's graph can have is \(3-1 = 2\).

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Most popular questions from this chapter

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. 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 maximum number of turning points to check whether it is drawn correctly. $$f(x)=-2 x^{3}(x-1)^{2}(x+5)$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the ZOOM OUT feature to show that f and g have identical end behavior. \(f(x)=x^{3}-6 x+1, g(x)=x^{3}\)

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. 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 maximum number of turning points to check whether it is drawn correctly. $$f(x)=-3(x-1)^{2}\left(x^{2}-4\right)$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{1-\frac{3}{x+2}}{1+\frac{1}{x-2}} $$

In Exercises \(98-99,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the \([\mathrm{ZOOMOUT}]\) feature to show that \(f\) and \(g\) have identical end behavior. $$f(x)=x^{3}-6 x+1, \quad g(x)=x^{3}$$
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