Question

write an equation for a rational function with: vertical asymptotes at x = -4 and x = 2 x intercepts at x = -6 and x = 1 horizontal asymptote at y = 4 y = add work next question

Explanation:

Step1: Determine the denominator

For vertical asymptotes at \( x = -4 \) and \( x = 2 \), the denominator of the rational function must have factors of \( (x + 4) \) and \( (x - 2) \). So the denominator is \( (x + 4)(x - 2) \).

Step2: Determine the numerator

For \( x \)-intercepts at \( x = -6 \) and \( x = 1 \), the numerator must have factors of \( (x + 6) \) and \( (x - 1) \). So the numerator is \( (x + 6)(x - 1) \).

Step3: Determine the leading coefficient

For the horizontal asymptote \( y = 4 \), the degrees of the numerator and denominator are the same (both are degree 2), so the leading coefficient of the numerator divided by the leading coefficient of the denominator must be 4. The leading coefficient of the numerator (from \( (x + 6)(x - 1)=x^2 + 5x - 6 \)) is 1, and the leading coefficient of the denominator (from \( (x + 4)(x - 2)=x^2 + 2x - 8 \)) is 1. So we multiply the numerator by 4 to get the leading coefficient ratio to be 4.
So the rational function is \( y=\frac{4(x + 6)(x - 1)}{(x + 4)(x - 2)} \)

Answer:

\( y=\frac{4(x + 6)(x - 1)}{(x + 4)(x - 2)} \) (or expanded form \( y=\frac{4x^2 + 20x - 24}{x^2 + 2x - 8} \))