To find the horizontal asymptote of the given rational function
[tex]\[ y = \frac{6x - 24}{2x + 12} \][/tex]
we need to compare the degrees of the numerator and the denominator.
1. Identify the degrees of the numerator and the denominator:
- The degree of the numerator [tex]\(6x - 24\)[/tex] is 1 (since the highest power of [tex]\(x\)[/tex] is [tex]\(x^1\)[/tex]).
- The degree of the denominator [tex]\(2x + 12\)[/tex] is also 1 (since the highest power of [tex]\(x\)[/tex] is [tex]\(x^1\)[/tex]).
2. Determine the horizontal asymptote based on the degrees:
- When the degrees of the numerator and the denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients (the coefficients of the highest degree terms).
3. Find the leading coefficients:
- The leading coefficient of the numerator [tex]\(6x\)[/tex] is 6.
- The leading coefficient of the denominator [tex]\(2x\)[/tex] is 2.
4. Calculate the horizontal asymptote:
- The horizontal asymptote is given by the ratio of the leading coefficients:
[tex]\[ y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} \][/tex]
[tex]\[ y = \frac{6}{2} = 3.0 \][/tex]
Therefore, the horizontal asymptote of the function [tex]\( y = \frac{6x - 24}{2x + 12} \)[/tex] is
[tex]\[ y = 3.0 \][/tex]
