To find the horizontal asymptote of the function [tex]\( y = \frac{3x + 27}{x - 9} \)[/tex], we need to look at the degrees of the polynomials in the numerator and the denominator.
1. Identify the degrees of the polynomials:
- The numerator is [tex]\( 3x + 27 \)[/tex]. The degree of the numerator is 1, since the highest power of [tex]\( x \)[/tex] is [tex]\( x^1 \)[/tex].
- The denominator is [tex]\( x - 9 \)[/tex]. The degree of the denominator is also 1, as the highest power of [tex]\( x \)[/tex] is [tex]\( x^1 \)[/tex].
2. Compare the degrees of the numerator and the denominator:
- Since the degrees of both the numerator and the denominator are equal, 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]\( 3x + 27 \)[/tex] is 3.
- The leading coefficient of the denominator [tex]\( x - 9 \)[/tex] is 1.
4. Calculate the horizontal asymptote:
- The horizontal asymptote is given by [tex]\( \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}} \)[/tex].
- Therefore, the horizontal asymptote is [tex]\( y = \frac{3}{1} = 3 \)[/tex].
So, the horizontal asymptote for the function [tex]\( y = \frac{3x + 27}{x - 9} \)[/tex] is [tex]\( y = 3 \)[/tex].
