Write an equation for a rational function given the characteristics.

Question

Question: Vertical asymptotes at x=−1 and x=3, x-intercepts at (−3,0) and (1,0), horizontal asymptote at y=−2

Answer 1

Vertical asymptotes make the denominator = 0.

(x + 1)(x - 3) = 0

x-intercepts make the numerator = 0

(x + 3)(x - 1) = 0

So far, we have ((x + 3)(x - 1))/((x + 1)(x - 3))

To find the horizontal asymptote, the leading degrees have to be the same but the leading coefficient/leading coefficient has to equal -2, aka -2/1

(-2(x + 3)(x - 1))/(x + 1)(x - 3))

f(x) = (-2x² - 4x + 6)/(x² - 2x - 3)

Hope this helps!

Answer 2

I love these types of questions. It's like trying to solve a murder mystery given a set of clues. So let's work through each of these clues and see where they lead us.

"Vertical asymptotes at x=−1 and x=3" tells us that we need (x+1)(x-3) in the denominator.

"x-intercepts at (−3,0) and (1,0)" tells us that we have (x+3)(x-1) in the numerator.

"horizontal asymptote at y=−2" tells us that the coefficient of highest degree in the numerator should be -2.

Piecing this all together, the function should be:

y = -2(x+3)(x-1) / ((x+1)(x-3))