SOLUTION:(2) Graph each of the following rational functions by using the asymptotes and plotting a few other points. 4 (a) y = ----- x - 2 1 (b) y = ----- x + 3 -4 (c) y = ----- x - 2 -1 (d) y = ----- x + 3
4
(a) y = -----
x - 2
x - 2 = 0 ==> x = 2
4/x 4/x
y = --------- = ------- ==> y = 0
x/x - 2/x 1 - 2/x
There is a vertical asymtote at x = 2 and a horizontal asymtote at y = 0.
1
(b) y = -----
x + 3
x + 3 = 0 ==> x = -3 (vertical asymtote)
1/x 1/x
y = --------- = ------- ==> y = 0 (horizontal asymtote)
x/x + 3/x 1 + 3/x
The asymtotes are x = -3 and y = 0.
-4
(c) y = -----
x - 2
x - 2 = 0 ==> x = 2
-4/x -4/x
y = --------- = ------- ==> y = 0
x/x - 2/x 1 - 2/x
There is a vertical asymtote at x = 2 and a horizontal asymtote at y = 0.
-1
(d) y = -----
x + 3
x + 3 = 0 ==> x = -3 (vertical asymtote)
-1/x -1/x
y = --------- = ------- ==> y = 0 (horizontal asymtote)
x/x + 3/x 1 + 3/x
The asymtotes are located at x = -3 and at y = 0.
