Write an equation for a rational function with: Vertical asymptotes at x = 4 and x = -6 x-intercepts at x = -5 and x = 3 y-intercept at 7

Question

Write an equation for a rational function with:

Vertical asymptotes at x = 4 and x = -6

x-intercepts at x = -5 and x = 3

y-intercept at 7

Answer 1

Hi Dayanara,

when you want to write rational function you need to make a fraction.

because of asymptotes denominator will be (x-4)(x+6)

because of x-int nominator will be (x+5)(x-3)

now you have f(x) = K(x+5)(x-3) /(x-4)(x+6)

you need to find K

because of y-int =7

plug in x=0 and f(0) =7 in above function and find k

7 = k(0+5)(0-3)/(0-4)(0+6) K=56/5

as a result f(x) = 56(x+5)(x-3) /5(x-4)(x+6)

I hope it is useful,

Minoo

Answer 2

WE loking for this function as f(x) =(a(x - 3)(x- (-5)))/((x - (-6))(x - 4)) = ((x - 3)(x + 5))/((x + 6)(x - 4)).

To get a we uzing y-intercept f(0) = 7

We have:a(-3)(5)/(6)(-4) = 7. From here a = 7·24/15 = 56/5.

Answer: f(x) = (56(x - 3)(x + 5))/(5(x + 6)(x - 4))