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Chapter 3: Problem 81

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?

Short Answer

Expert verified
A third-degree polynomial function with a negative leading coefficient is not suitable for modeling non-negative real-world phenomena over a long period of time since the function will eventually fall below zero while the phenomena never do.

Step by step solution

01

Identifying Characteristics of a Third-Degree Polynomial Function

A third-degree polynomial function, also known as a cubic function, has a characteristic shape that changes direction twice, giving the graph up to three x-intercepts (roots) and creating two turning points. The leading coefficient of a cubic function determines the end behavior of the graph. In this case, a negative leading coefficient in a third-degree polynomial function means that as x approaches positive infinity, y will approach negative infinity, and as x approaches negative infinity, y will approach positive infinity. It is crucial to understand this characteristic to comprehend why this is not appropriate for modeling non negative real-world phenomena.
02

Understanding Non-Negative Real-World Phenomena

Non-negative real-world phenomena refer to scenarios or events that never fall below zero. The count of a population, amount of money, or physical measurements like height or weight are some examples. For example, the population of a certain area, regardless of how it fluctuates, cannot go below zero.
03

Linking the Third-Degree Polynomial Function to Real-World Phenomena

Relating these concepts, a third-degree polynomial function with a negative leading coefficient, over a long period of time (as x approaches positive infinity), will ultimately fall below zero (y approaches negative infinity), which contradicts the nature of non-negative real-world phenomena.

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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. $$f(x)=(x+3)(x+1)^{3}(x+4)$$

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. $$f(x)=\frac{1}{2}-\frac{1}{2} x^{4}$$

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. $$f(x)=-3(x-1)^{2}\left(x^{2}-4\right)$$

I graphed \(f(x)=(x+2)^{3}(x-4)^{2},\) and the graph touched the \(x\)-axis and turned around at \(-2\)

Exercises 113鈥�115 will help you prepare for the material covered in the next section. Use $$ \frac{2 x^{3}-3 x^{2}-11 x+6}{x-3}=2 x^{2}+3 x-2 $$ to factor \(2 x^{3}-3 x^{2}-11 x+6\) completely
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