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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

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 \( \text { of } f(x)=\frac{1}{x} \text { to graph } g \). $$ g(x)=\frac{2 x-9}{x-4} $$

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose \(y\) -coordinate is the same as the given point. $$ f(x)=3(x+2)^{2}-5 ;(-1,-2) $$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$ f(x)=(x-4)^{2}-1 $$

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

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?
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