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Chapter 3: Problem 95

Use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-2 x^{3}+6 x^{2}+3 x-1$$

Short Answer

Expert verified
The graph of the function \(f(x)=-2 x^{3}+6 x^{2}+3 x-1\) starts in quadrant 2 and ends in quadrant 4 due to its negative leading coefficient and odd degree.

Step by step solution

01

Identify the Leading Coefficient and Degree of Polynomial

The degree of the given polynomial function \(f(x)=-2 x^{3}+6 x^{2}+3 x-1\) is 3 (because the highest exponent is 3) and the leading coefficient is -2.
02

Determine the End Behavior

Based on the degree and the leading coefficient, we can determine the end behavior of the polynomial. Since the degree is odd and the leading coefficient is negative, as \(x\) approaches \(+\infty\), \(f(x)\) will approach \(-\infty\), and as \(x\) approaches \(-\infty\), \(f(x)\) will approach \(+\infty\). This implies that the graph of \(f(x)\) will start in quadrant 2 and end in quadrant 4.
03

Graph the Polynomial Function

Using a graphing utility, input the function \(f(x)=-2 x^{3}+6 x^{2}+3 x-1\). Make sure to adjust the viewing rectangle to show the end behavior of the graph. The graph will display the cubic function starting from quadrant 2 and extending to quadrant 4.

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