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Chapter 2: Problem 97

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

Short Answer

Expert verified
The end behavior of function \(f(x)=-x^{5}+5 x^{4}-6 x^{3}+2 x+20\) can be observed in the graph with the left end rising and the right end falling.

Step by step solution

01

Look at the highest degree term

First, identify the highest degree term in the given polynomial. For the function \(f(x)=-x^{5}+5 x^{4}-6 x^{3}+2 x+20\), the highest degree term is \(-x^{5}\).
02

Determine the end behavior

After identifying the highest degree term, make a conclusion about the end behavior. A fifth degree polynomial with a negative leading coefficient like \(-x^{5}\) means as \(x\) approaches infinity, \(y\) approaches negative infinity (\(x \to \infty, y \to -\infty\)) and as \(x\) approaches negative infinity, \(y\) approaches positive infinity (\(x \to -\infty, y \to \infty\)). This shows that the left end of the graph rises and the right end of the graph falls.
03

Graph the Polynomial

Enter the function \(f(x)=-x^{5}+5 x^{4}-6 x^{3}+2 x+20\) into a graphing utility and choose a large viewing rectangle to best portray the end behavior. The graph of this function shows a curve that rises to the left and falls to the right, which confirms the observation made about the end behavior in Step 2.

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