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Chapter 3: Problem 79

What is meant by the end behavior of a polynomial function?

Short Answer

Expert verified
The end behavior of a polynomial function refers to the value that the function approaches as x moves towards positive or negative infinity. It is determined by the degree and leading coefficient of the function. If the degree is even, both ends of the graph will approach the same direction (both going up if the leading coefficient is positive, both going down if negative). If the degree is odd, one end will go up and the other will go down, dependent on the sign of the leading coefficient.

Step by step solution

01

Understanding Degree and Leading Coefficient

For any polynomial function represented as \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \( a_n, a_{n-1}, ..., a_0 \) are constants, the coefficient \(a_n\) of the highest power of 'x' is called the 'leading coefficient'. The highest power 'n' is referred to as the 'degree'.
02

Rules of End behavior

For a polynomial function, if the 'degree' is even, as 'x' approaches infinity or negative infinity, 'f(x)' will approach positive infinity if the 'leading coefficient' is positive. If the leading coefficient is negative, 'f(x)' will approach negative infinity. If the 'degree' is odd, as 'x' approaches infinity or negative infinity, 'f(x)' will approach different directions based on the sign of the leading coefficient.
03

Summary

In summary, the end behavior of a polynomial function refers to the value that the function approaches as 'x' moves towards positive or negative infinity, and this is determined by the degree and leading coefficient of the polynomial function.

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Most popular questions from this chapter

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has a vertical asymptote given by x=1, a slant asymptote whose equation is y=x, y -intercept at 2, and x -intercepts at -1 and 2.

Solve and graph the solution set on a number line: $$\frac{2 x-3}{4} \geq \frac{3 x}{4}+\frac{1}{2}$$ (Section 1.7, Example 5)

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. $$f(x)=6 x-x^{3}-x^{5}$$

Why is a third-degree polynomial function with a negative leading coefficient not appropriate for modeling nonnegative real-world phenomena over a long period of time?

In Exercises 94鈥�97, use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-2 x^{3}+6 x^{2}+3 x-1$$
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