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Chapter 2: Problem 21

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 function \(f(x)=5 x^{4}+7 x^{2}-x+9\) is an increase as \(x\) approaches both positive and negative infinity, according to the leading coefficient test.

Step by step solution

01

Identify the leading term

First, let's establish that the leading term in the function is the one with the highest exponent. In this function, \(f(x)=5 x^{4}+7 x^{2}-x+9\), that term is \(5x^4\). So, the leading coefficient is 5, and the degree of the polynomial is 4.
02

Apply the leading coefficient test

The leading coefficient test tells us that if the degree of the polynomial is even, like in this case, then the end behavior of the function will be the same on either end. If the coefficient is positive, the function will rise to the right and left (increase as \(x\) approaches \(\pm \infty\)). As the leading coefficient in this function is positive, we know that the function will rise to the right and the left.
03

State the end behavior of the function

Given our previous steps, we can now state that as \(x\) approaches positive or negative infinity, the function \(f(x)=5 x^{4}+7 x^{2}-x+9\) will also approach infinity. This describes the function's end behavior.

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