Question

1.Use completing the square to describe the graph of the following function. Support your answer graphically.
f(x) = -2x^2 + 4x + 5

2. Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function. R(x) = 5x/x+7
The vertical asymptote(s) is/are x =
The horizontal asymptote(s) is/are y =
The oblique asymptote(s) is/are y =

Answer 1

f(x) = -2x^2 + 4x + 5
= -2(x^2 - 2x) + 5
= -2(x^2 - 2x + 1) + 5 - -2(1)
= -2(x-1)^2 + 7

this is a parabola with vertex at (1,7), opening downwards, with roots at 1 +/- sqrt(14)/2

5x/(x+7) = 5 - 35/(x+7)

has a vertical asymptote at x=-7
and a horizontal asymptote at y=5
no oblique asymptote, since the degree of top and bottom are the same.