To find the horizontal asymptote of the rational function
[tex]\[
y = \frac{4x - 12}{x + 2}
\][/tex]
we need to consider the degrees of the polynomial in the numerator and the polynomial in the denominator.
1. Identify the degrees of the polynomials:
- The degree of the numerator (4x - 12) is 1 because the highest power of x is 1.
- The degree of the denominator (x + 2) is also 1 because the highest power of x is 1.
2. Compare the degrees of the numerator and the denominator:
- Since the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator.
3. Find the leading coefficients:
- The leading coefficient of the numerator (4x - 12) is 4.
- The leading coefficient of the denominator (x + 2) is 1.
4. Calculate the horizontal asymptote:
- The horizontal asymptote is given by the ratio of the leading coefficients:
[tex]\[
y = \frac{4}{1}
\][/tex]
- Simplifying this gives:
[tex]\[
y = 4
\][/tex]
Hence, the horizontal asymptote of the function
[tex]\[
y = \frac{4x - 12}{x + 2}
\][/tex]
is
[tex]\[
y = 4.
\][/tex]
