Write an equation for a rational function with: Vertical asymptotes at x = 3 and x = -2 x intercepts at x = -3 and x = 4 y intercept at 3

Question

Write an equation for a rational function with:

Vertical asymptotes at x = 3 and x = -2

x intercepts at x = -3 and x = 4

y intercept at 3

Answer 1

A vertical asymptote implies a zero in the denominator. So, (x +2) and (x-3) are factors in the denominator.

An x- intercept means y = 0. In a fraction that only occures when the numerator is 0. So, (x + 3) and (x - 4) must be factors in the numerator.

So far, we have y = K((x+3)(x -4)) / ((x+2)(x - 3)) where K is any multiple of the function with thw same VA and intercepts.

if the y intercept is at 3, the function must contain the point (0, 3)

Substituting x = 0, and y = 3 will allow us to solve for K to find the specific equation satisfying all conditions.

3 = K((3)(-4))/((2)(-3))

3 = K(-12/-6)

3 = 2K

K = 3/2

S0 f(x) = (3 (x+3)(x-4))/ (2(x + 2)(x -3))