The graph of \(y = \frac{p(x)}{q(x)}\) is shown below, where p(x) and q(x) are quadratic. (Assume that the grid lines are at integers.) The horizontal asymptote is y = 2, and the only vertical asymptote is x = -2. Find \( \frac{p(3)}{q(3)}.\)
+128406 Since we have a vertical asymptote at x = -2.....then (x + 2) must be a factor of q(x)
Since we have a horizontal asymptote at y = 2, then 2 must be a factor of p(x) since p(x) and q(x) have the same degree
Since we have a hole at x = 5....then (x - 5) must be a factor of p(x) and q(x)
Since the point (1,0) is on the graph, then ( x - 1) must also be a factor of p(x)
So.....the function is
2(x - 5) (x - 1)
___________
(x - 5) ( x + 2)
And
p(3) 2(3 - 5)(3 - 1) 2 (-2)(2) -8 4
____ = ____________ = ________ = ___ = ___
q (3) (3 - 5) (3 + 2) (-2) (5) -10 5
So (3, 4/5) is on the graph
Here's a pic : https://www.desmos.com/calculator/gydybu0wek
