+170 The graph of \(y = \frac{p(x)}{q(x)}\) is shown below, where \(p(x)\) is linear and \(q(x)\) is quadratic. (Assume that the grid lines are at integers.)
Find \(\frac{p(-1)}{q(-1)}.\)
+128460 Since x = -3 and x = 2 are vertical asymptotes, then we might guess that the denominator might be of the form...
a(x + 3) (x - 2)
Since (0,0) is on the graph, then the numerator must be of the form.... bx
And the points (3,1) and (-2, 1) are on the graph
So...we have that
b (3) 3b
____________ = 1 ⇒ ________ = 1 ⇒ 3b = 6a ⇒ b = 2a
a (3 + 3)(3 - 2 ) 6a
And
b(-2) -2b
______________ = 1 ⇒ ________ = 1 ⇒ -2b = -4a ⇒ -2(2a) = -4a ⇒ a = 1
a(-2 + 3) (-2 - 2) a(1)(-4)
So p = bx = 2(a)x = 2(1)x = 2x
And q = 1 (x + 3) (x - 2) = (x + 3) (x - 2)
So p(-1 ) 2(-1) -2 -1 1
____ = ____________ = ________ = ___ = ___
q (-1) (-1 + 3) (-1 - 2) (2) (-3) -3 3
