Problem 80 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The end behavior of a polynomial function can be determined from its degree and leading coefficient using the Leading Coefficient Test. If the degree of the function is even, the end behavior of the function will be the same on both ends (either up or down), depending on whether the leading coefficient is positive or negative. If the degree is odd, the end behavior of the function will be opposite on both ends (up on one end and down on the other), also depending on the sign of the leading coefficient.

Step 1: Identify the Degree of the Polynomial

Look at the polynomial function and identify the term with the highest power of the variable. The exponent associated with this term is known as the degree of the polynomial. This degree can be either even or odd.

Step 2: Determine the Leading Coefficient

The coefficient of the term with the highest degree identified in Step 1 is the leading coefficient. This value can be either positive or negative.

Step 3: Apply the Leading Coefficient Test

Now, using both the degree of the polynomial (even/odd) and the sign of the leading coefficient (positive/negative), apply the Leading Coefficient Test as follows: 1. If the degree is even, and the leading coefficient is positive, the end behavior of the function is (\( y \to \infty \) as \( x \to \infty \)) and (\( y \to \infty \) as \( x \to -\infty \)).2. If the degree is even, but the leading coefficient is negative, the end behavior of the function is (\( y \to -\infty \) as \( x \to \infty \)) and (\( y \to -\infty \) as \( x \to -\infty \)).3. If the degree is odd, and the leading coefficient is positive, then the end behavior of the function is (\( y \to -\infty \) as \( x \to -\infty \)) and (\( y \to \infty \) as \( x \to \infty \)).4. For odd-degree functions, if the leading coefficient is negative, the end behavior of the function is (\( y \to \infty \) as \( x \to -\infty \)) and (\( y \to -\infty \) as \( x \to \infty \)).