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Chapter 3: Problem 96

Use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-x^{4}+8 x^{3}+4 x^{2}+2$$

Short Answer

Expert verified
The graph of the function \(f(x) = -x^{4} + 8x^{3} + 4x^{2} + 2\) starts from the top left (positive infinity), turns at points (0, 2) and (2, 10), and ends towards bottom right (negative infinity).

Step by step solution

01

Define the Function

The polynomial given is \(f(x)=-x^{4}+8 x^{3}+4 x^{2}+2\). The degree of this polynomial is 4, which means it will have a few changes in direction if we look at its graph.
02

Identify Key Points

Find the derivative of the function to locate any critical points or inflection points. The derivative of the function is \(f'(x)=-4x^{3}+24x^{2}+8x\). Set this to zero to find the x-values where the graph changes direction. Solving the equation \(-4x^3 + 24x^2 + 8x = 0\) for x gives the roots as \(x=0\) and \(x=2\), which would be the turning points of the graph.
03

Determine End Behavior

As the degree of the polynomial is even and the leading coefficient is negative, then the function will rise to the left and fall to the right. Therefore, the graph starts from positive infinity when \(x \to -\infty\) and decrease to negative infinity when \(x \to \infty\). This determines the end behavior of the function.
04

Sketching the Graph

Using the key points and end behavior deduced above, you can sketch the graph. The graph should pass through the points (0, 2) and (2, 10) (points obtained from substituting the solutions for x obtained at step 2 into the function), and displayed end behavior of starting from the top left (positive infinity), turning at (0, 2) , turning at (2, 10) and finally ending towards bottom right (negative infinity).

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