To find the horizontal asymptote of the rational function [tex]\(y = \frac{6x - 18}{x + 9}\)[/tex], follow these steps:
1. Identify the degree of the numerator and the denominator:
- The numerator is [tex]\(6x - 18\)[/tex], which is a polynomial of degree 1.
- The denominator is [tex]\(x + 9\)[/tex], which is also a polynomial of degree 1.
2. Determine the leading terms of the numerator and the denominator:
- The leading term in the numerator is [tex]\(6x\)[/tex].
- The leading term in the denominator is [tex]\(x\)[/tex].
3. Divide the leading coefficients of the highest degree terms:
- The coefficient of [tex]\(x\)[/tex] in the numerator is [tex]\(6\)[/tex].
- The coefficient of [tex]\(x\)[/tex] in the denominator is [tex]\(1\)[/tex].
4. Find the horizontal asymptote by dividing these coefficients:
[tex]\[
\text{Horizontal Asymptote} = \frac{\text{Numerator's leading coefficient}}{\text{Denominator's leading coefficient}} = \frac{6}{1} = 6
\][/tex]
So, the horizontal asymptote of the function [tex]\(y = \frac{6x - 18}{x + 9}\)[/tex] is [tex]\(y = 6\)[/tex].
