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Chapter 3: Problem 22

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=11 x^{4}-6 x^{2}+x+3$$

Short Answer

Expert verified
The end behavior of the polynomial \(f(x)=11x^{4}-6x^{2}+x+3\) is that it rises as \(x\) approaches positive infinity and also rises as \(x\) approaches negative infinity.

Step by step solution

01

Identify the degree and the leading coefficient

The degree of the polynomial \(f(x)=11x^{4}-6x^{2}+x+3\) is 4, which is the highest power of \(x\), and the leading coefficient is 11, which is the coefficient of the term with the highest degree, \(11x^{4}\).
02

Apply the Leading Coefficient Test

The Leading Coefficient Test states that if the degree is even and the leading coefficient is positive, the graph will rise to the right and rise to the left. Since the degree of the polynomial in this case is even (4) and the leading coefficient is positive (11), the graph of the function will rise as \(x\) approaches positive infinity and will also rise as \(x\) approaches negative infinity. This describes the end behavior of the polynomial.

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