To find the horizontal asymptote of the rational function [tex]\( y = \frac{4x + 32}{x - 8} \)[/tex], follow these steps:
1. Identify the degrees of the numerator and the denominator:
The highest degree term in the numerator is [tex]\(4x\)[/tex], which is degree 1. The highest degree term in the denominator is [tex]\(x\)[/tex], which is also degree 1.
2. Compare the degrees:
Both the numerator and the denominator have the same degree, which is 1.
3. Determine the horizontal asymptote:
When the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients.
- The leading coefficient of the numerator ([tex]\(4x\)[/tex]) is 4.
- The leading coefficient of the denominator ([tex]\(x\)[/tex]) is 1.
Therefore, the horizontal asymptote [tex]\(y\)[/tex] is given by:
[tex]\[
y = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}} = \frac{4}{1} = 4
\][/tex]
So, the horizontal asymptote of the function [tex]\( y = \frac{4x + 32}{x - 8} \)[/tex] is [tex]\( y = 4 \)[/tex].
