Question

section 3.7
score: 7/10 answered: 7/10
question 8
write an equation for a rational function with:
vertical asymptotes at x = -3 and x = 6
x intercepts at x = 4 and x = -2
horizontal asymptote at y = 8
y =
question help:

Explanation:

Step1: Find the denominator

For vertical asymptotes at \(x = - 3\) and \(x = 6\), the denominator \(Q(x)\) is \((x + 3)(x - 6)=x^{2}-3x - 18\) since vertical asymptotes occur where the denominator is zero (and numerator is non - zero).

Step2: Find the numerator

For \(x\) - intercepts at \(x = 4\) and \(x=-2\), the numerator \(P(x)\) is \(a(x - 4)(x + 2)=a(x^{2}-2x - 8)\) since \(x\) - intercepts occur where the numerator is zero (and denominator is non - zero).

Step3: Determine the leading coefficient \(a\)

The horizontal asymptote of a rational function \(y=\frac{P(x)}{Q(x)}\) (where \(P(x)\) and \(Q(x)\) are polynomials of degree \(n\) and \(m\) respectively) is given by \(y = 0\) if \(n\lt m\), \(y=\frac{\text{leading coefficient of }P(x)}{\text{leading coefficient of }Q(x)}\) if \(n = m\), and no horizontal asymptote if \(n\gt m\).
Here, since the horizontal asymptote \(y = 8\) and the degree of the numerator and denominator (both degree \(2\)) are equal. The leading coefficient of the numerator is \(a\) and the leading coefficient of the denominator is \(1\). So, \(\frac{a}{1}=8\), which means \(a = 8\).

Answer:

\(y=\frac{8(x - 4)(x + 2)}{(x + 3)(x - 6)}=\frac{8x^{2}-16x - 64}{x^{2}-3x - 18}\)