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Chapter 3: 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 graph of the polynomial function \(f(x)=-5x^4 +7x^2 -x+9\) falls to the left and right.

Step by step solution

01

Identifying the degree and sign of leading coefficient

To utilize the Leading Coefficient Test, first identify the degree and the sign of the leading coefficient of the polynomial function. The provided function is \(f(x) =-5 x^{4} +7 x^{2} -x+9\). The degree of the polynomial function is 4 and the leading coefficient is -5.
02

Apply Leading Coefficient Test

The Leading Coefficient Test states if the degree of the polynomial is even and the leading coefficient is positive, the end behavior of the graph will rise to the left and right. If the degree of the polynomial is even and the leading coefficient is negative, the end behavior of the graph will fall to the left and right. If the degree of the polynomial is odd and leading coefficient is positive, the graph will fall to the left, and rise to the right. If the degree of the polynomial is odd and the leading coefficient is negative, the graph will rise to the left and fall to the right.
03

Determine end behavior

From step 1, the degree of the provided function is 4 (even) and the leading coefficient is -5 (negative). So according to the Leading Coefficient Test derived in Step 2, since degree is even and the leading coefficient is negative, the graph falls to the left and right.

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