Unsure of how to write an equation for rational functions

Question

Write an equation for a rational function with:

Vertical asymptotes at x = -4 and x = 1

x intercepts at x = -6 and x = 5

y intercept at 2

 

y=

Answer 1

Let y = F(X)

 

If there are vertical asymptotes, this means there are no x values at those points.

 

y = g(x)/((x+-4)(x-1)), where g(x) is a polynomial since this is a rational function.

 

X-intercepts are the points (x, 0)

In this case (-6, 0) and (5, 0) are x intercepts.

 

Then 0 = g(=6)/14 => g(-6) = 0

and 0 = g(5)/36 => g(5) = 0

Since g(-6) = g(4) = 0 and this is polynomial, g(x) = k(x+6)(x-5) where k

is a constant.

 

y = k(x+6)(x-5)/((x +4)(x-1))

 

Finally, the y intercept at 2 is the point (0,2)

2 = k(6)(-5)/(4*-1) = 15k/2

k = 4/15

 

The rational function is y = (4/15)(x+6)(x-5)/((x+4)(x-1))