Question

Write an equation of a rational function that satisfies all of these conditions below. Show how each condition is satisfied in order to get full marks.
- y-intercept of -5
- x-intercepts at x=-1 and x=5
- Hole at x=-2
- Horizontal asymptote at y=6
- No vertical asymptotes

Answer 1

Hole at x=-2 ||| y = (x+2)/(x+2)
x-intercepts at x=-1 and x=5 ||| y = (x+2)(x+1)(x-5)/(x+2)
Horizontal asymptote at y=6 ||| y = 6(x+2)(x+1)(x-5) / (x+2)(x^2+1)
now, y(0) = -30
so, divide by -6 at x=0 and you get
y = 6(x+2)(x+1)(x-5) / (x+2)(x^2-6)

see the graph at

https://www.wolframalpha.com/input/?i=%286+%28x%2B2%29%28x%2B1%29%28x-5%29%29+%2F+%28%28x%2B2%29%28x%5E2-6%29%29+