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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 \( \frac{p(3)}{q(3)}.\)

 Jul 28, 2019
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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

 

cool cool cool

 Jul 28, 2019
edited by CPhill  Jul 28, 2019

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