precalculus help

Question

Write an equation for a rational function with:

Vertical asymptotes at x = 6 and x = -4

x-intercepts at x = -5 and x = 3

y-intercept at 6

y = 

Answer 1

If we have vertical asymptotes at x=6 and x=-4, then the denominator will be (x-6)(x+4)

If our x-intercepts are x=-5 and x=3, then the numerator will be (x+5)(x-3)

So far, we have y = [(x+5)(x-3)]/[(x-6)(x+4)] as our equation. To account for a y-intercept of y=6, we need to have a constant "a" multiplying the function and then we would solve for "a" using our given y-intercept:

y = a[(x+5)(x-3)]/[(x-6)(x+4)]

6 = a[(0+5)(0-3)]/[(0-6)(0+4)]

6 = a[(5)(-3)]/[(-6)(4)]

6 = -15a/(-24)

6 = 5a/8

48 = 5a

48/5 = a

Thus, our final equation that meets all the criteria is y = 48[(x+5)(x+3)]/[5(x-6)(x+4)].

I hope this helped! Feel free to comment below if you need me to clarify anything!