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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 \(25-32,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+5 x^{2}-9 x-45$$

Use long division to rewrite the equation for \(g\) in the form $$ \text {quotient}+\frac{\text {remainder}}{\text {divisor}} $$ Then use this form of the function's equation and transformations \( \text { of } f(x)=\frac{1}{x} \text { to graph } g \). $$ g(x)=\frac{2 x-9}{x-4} $$

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

Can the graph of a polynomial function have no x-intercepts? Explain.

Use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. \(f(x)=-x^{5}+5 x^{4}-6 x^{3}+2 x+20\)
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