Question

Identify the rational function whose graph is given below. The y-intercept is (0,−1/2).

Answer 1

The numerator relates to the x-intercepts:

(-1,0) means (x+1) is a factor in the numerator.

The denominator relates to the vertical intercepts:

x = -2 as a VA means (x+2) is a factor in the denominator.

x = 1 as a VA means (x-1) is a factor in the denominator.

Since the x-intercept crosses and the VA's are "one up, one down", we know these all have odd multiplicities.

So, we think the function could be
`r(x) = (x+1)/((x+2)(x-1))`

Checking the y-intercept with this function

`r(0)= (0+1)/((0+2)(0-1)) = (1)/(-2)=-(1)/(2)`

This checks out. That's your function:
`r(x) = (x+1)/((x+2)(x-1))`