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Chapter 3: Problem 59

Explain how to use the Leading Coefficient Test to determine the end behavior of a polynomial function.

Short Answer

Expert verified
The end behavior of a polynomial function can be determined using the Leading Coefficient Test. The end behavior is dependent on the degree of the polynomial and the sign of its leading coefficient.

Step by step solution

01

Understand Terms

Before tackling the problem, it's essential to understand what the terms mean. In a polynomial, the leading coefficient is the coefficient of the term of highest degree. The degree of the polynomial is the highest power in the polynomial.
02

Leading Coefficient Test Organization

The Leading Coefficient Test is generally organized as follows : \n- If the degree of the polynomial is odd and the leading coefficient is positive, the end behavior is from the third quadrant to the first quadrant.- If the degree of the polynomial is odd and the leading coefficient is negative, the end behavior goes from the second quadrant to the fourth quadrant.- If the degree of the polynomial is even and the leading coefficient is positive, the end behavior is from the second quadrant to the first quadrant.- If the degree of the polynomial is even and the leading coefficient is negative, the end behavior goes from the third quadrant to the fourth quadrant.
03

Determine End Behavior

Using the above information from the Leading Coefficient Test, one can determine the end behavior of any polynomial function by simply identifying the highest degree term in the polynomial and its leading coefficient. Then, based on whether the degree is odd or even, and whether the leading coefficient is positive or negative, apply the established relationship to decipher the end behavior of the function.

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Most popular questions from this chapter

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if any, of the functions graph.

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=\frac{x^{2}+7}{3}$$

Use a graphing utility to graph \(y=\frac{1}{x}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x$$

In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=x^{3}+13 x^{2}+10 x-4$$
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