To find the horizontal asymptote of the rational function [tex]\( y = \frac{3x + 12}{x - 6} \)[/tex], we need to analyze its behavior as [tex]\( x \)[/tex] approaches infinity or negative infinity. The horizontal asymptote describes the value that [tex]\( y \)[/tex] approaches as [tex]\( x \)[/tex] becomes very large in magnitude (either positively or negatively).
Here’s a step-by-step method to determine the horizontal asymptote:
1. Identify the degrees of the numerator and the denominator:
- The numerator is [tex]\( 3x + 12 \)[/tex].
- The denominator is [tex]\( x - 6 \)[/tex].
- Both the numerator and denominator are polynomials of degree 1.
2. Compare the degrees of the polynomials:
- Since the degrees of the numerator and denominator are the same (both are degree 1), the horizontal asymptote is found by taking the ratio of the leading coefficients.
3. Determine the leading coefficients:
- The leading coefficient of the numerator [tex]\( 3x \)[/tex] is [tex]\( 3 \)[/tex].
- The leading coefficient of the denominator [tex]\( x \)[/tex] is [tex]\( 1 \)[/tex].
4. Calculate the horizontal asymptote:
- The horizontal asymptote is given by the ratio of the leading coefficients of the numerator and the denominator:
[tex]\[ \text{Horizontal Asymptote} = \frac{3}{1} = 3 \][/tex]
Therefore, the horizontal asymptote of the function [tex]\( y = \frac{3x + 12}{x - 6} \)[/tex] is [tex]\( y = 3 \)[/tex].
This means that as [tex]\( x \)[/tex] approaches infinity or negative infinity, the value of [tex]\( y \)[/tex] approaches 3.
