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Chapter 3: 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
To graph the function, input it into a graphing utility set to a wide viewing window to see the end behavior. The graph results from the function: \(f(x)=-x^{5}+5 x^{4}-6 x^{3}+2 x+20\). The graph opens downwards, exits the graph on opposite ends due to the odd power, and follows the predicted end behavior. The end behavior is: as \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity, and as \(x\) approaches negative infinity, \(f(x)\) approaches positive infinity.

Step by step solution

01

Identify the function

The given function is \(f(x)=-x^{5}+5 x^{4}-6 x^{3}+2 x+20\). This is a polynomial of degree 5.
02

Consider the leading term

Notice that the leading term of the polynomial is \(-x^{5}\). The negative sign means the function opens downwards. Since the power is odd, the function will exit the graph on opposite sides.
03

Use a graphing utility

Input the function into the graphing utility. Set a wide viewing window to observe the function across a broad range of x-values.
04

Check the End Behavior

Observe the graph. The long-term behavior of the function (i.e., as \(x\) approaches positive or negative infinity) is significant. Because the degree of the polynomial is odd and the leading coefficient is negative, as \(x\) approaches positive infinity, \(f(x)\) should approach negative infinity, and as \(x\) approaches negative infinity, \(f(x)\) should approach positive infinity. The graph's behavior should align with these predictions.

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