Question

The x-intercepts are marked with points located at (5,0) and (-6,0), while the y-intercept is marked with point located at (0,3/2). The asymptotes are y=1, x=-5, and x=4. Give your formula as a reduced rational function.

Answer 1

The horizontal asymptote at y=1 indicates that the numerator and denominator are of the same degree.

The vertical asymptotes indicate that
y = p(x) / (x+5)(x-4)
for some quadratic polynomial p(x)

The x-intercepts then dictate that
y = a(x+6)(x-5) / (x+5)(x-4)

The y-intercept at (0,3/2) means that
a(6)(-5) / (5)(-4) = 3/2
a = 3/2 * 20/30 = 1

So, y = (x+6)(x-5) / (x+5)(x-4) = (x^2+x-30)/(x^2+x-20)

See the graph at

https://www.wolframalpha.com/input/?i=(x%5E2%2Bx-30)%2F(x%5E2%2Bx-20)+for+-7+%3C%3D+x+%3C%3D+6