Question

A rational function, R(x) has the following characteristics:

a vertical asymptote at x = 3,
a horizontal asymptote at y = 2,
and a hole at (2, −2).

Sketch the function and determine what it could be using the following steps:

Put in the factor that would account for the vertical asymptote at x = 3.
Add in the factors that would account for a hole at x = 2.
Determine what must be true about the numerator and denominator for there to be a horizontal asymptote at y = 2.
Add the factors that would account for the horizontal asymptote at y = 2.
Describe what you must do in order for the hole to appear at (2, −2).
Write the completed function.

Answer 1

vertical asymptote at x = 3
y = a/(x-3)

horizontal asymptote at y = 2
y = 2x/(x-3)
y(2) = 4/-1 = -4
so let's use
y = 2(x-1)/(x-3)
y(2) = 2/-1 = -2

a hole at (2, −2)
y = 2(x-1)(x-2) / (x-2)(x-3)