Explain how to use end behaviour to find the equation of the horizontal asymptote of a rational function.
To use end behavior to find the equation of the horizontal asymptote of a rational function, follow these steps:
Step 1: Determine the degree of the numerator and denominator of the rational function. The degree is the highest power of the variable in the polynomial.
Step 2: Compare the degrees of the numerator and denominator to identify the end behavior of the rational function:
- If the degree of the numerator is less than the degree of the denominator, the function has a horizontal asymptote at y = 0 (the x-axis).
- If the degrees of the numerator and denominator are equal, the function has a horizontal asymptote at the ratio of the leading coefficients of the numerator and denominator.
- If the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote. Instead, it will have a slant asymptote, where the equation of the slant is found using polynomial long division.
Let's go through an example to illustrate this:
Example: Find the equation of the horizontal asymptote of the rational function f(x) = (3x^2 - 5x + 2) / (x^2 - 2x + 1).
Step 1: Determine the degree of the numerator and denominator. In this case, both the numerator and denominator have a degree of 2.
Step 2: Since the degrees of the numerator and denominator are equal, the function has a horizontal asymptote at the ratio of the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. Therefore, the equation of the horizontal asymptote is y = 3/1, which simplifies to y = 3.
So, the equation of the horizontal asymptote for f(x) = (3x^2 - 5x + 2) / (x^2 - 2x + 1) is y = 3.