Clifland

2021-08-17

Whether the given function is an appropriate model for extended years and explain the reason.
Given: The polynomial function is $f\left(x\right)=-0.87{x}^{3}+0.35{x}^{2}+81.62x+7684.94$
Here, x represents the number of years after 1987 and f(x) represents the number of thefts in that respective year.

Latisha Oneil

Procedure used:
As x increases or decreases without bound, the graph of the polynomial function.
$f\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\cdots +{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}\left({a}_{n}\ne q0\right)$ eventually rises or falls. In particular,
(1) For n odd:
If the leading coefficient is positive, the graph falls to the left and rises to the right.
If the leading coefficient is negative, the graph rises to the left and falls to the right.
(2) For n even:
If the leading coefficient is positive, the graph rises to the left and rises to the right.
If the leading coefficient is negative, the graph falls to the left and falls to the right.
Description:
The degree of the given function is observed to be 3, which is odd.
The odd polynomial function have opposite behavior at each end.
The number ${a}_{n}$, which is the coefficient of the variable with highest power is called the leading coefficient.
The coefficient ${x}^{3}$ is noticed to be -0.87. That is, the leading coefficient is negative.
Thus, by above procedure, the graph of $f\left(x\right)=-0.87{x}^{3}+0.35{x}^{2}+81.62x+7684.94$ rises to the left and falls to the right.
This implies that the number of thefts in x years after 1987 will be negative as the number the years increases.
This is impossible. So, it is not capable for modeling the number of thefts in United States for the extended years.

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