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Chapter 2: Problem 23

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-5 x^{4}+7 x^{2}-x+9$$

Short Answer

Expert verified
The end behavior of the polynomial function \(f(x)=-5x^{4}+7x^{2}-x+9\) is: as \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity, and as \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity. Therefore, the ends of the graph point downward.

Step by step solution

01

Identify the Leading Term

First, locate the leading term in the polynomial, which is the term with the highest exponent. In this polynomial, \(f(x)=-5x^{4}+7x^{2}-x+9\), the leading term is -5x^{4}, where -5 is the leading coefficient and 4 is the degree of the polynomial.
02

Apply the Leading Coefficient Test

Apply the Leading Coefficient Test using the identified leading term. Since the degree (4) is even and the leading coefficient (-5) is negative, the test tells us that as x approaches both positive or negative infinity, the graph descends to negative infinity. In other words, the ends of the graph point downward.
03

Conclude the End Behavior

Based on the Leading Coefficient Test, we can conclude that as x approaches positive infinity, \(f(x)\) approaches negative infinity, and as x approaches negative infinity, \(f(x)\) also approaches negative infinity.

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