SOLUTION:(3) Graph each of the following rational functions by using the intercepts and asymptotes. x (a) y = ------ 2x - 4 3x (b) y = ------ x - 3 2x - 4 (c) y = ------- x 1 - x (d) y = ------- x
x
(a) y = ------
2x - 4
2x - 4 = 0 ==> 2x = 4 ==> x = 2 (vertical asymtote)
x/x 1
y = ---------- = ------- ==> y = 1/2 (horizontal asymtote)
2x/x - 4/x 2 - 4/x
y-intercept is when x = 0 ==> y = 0
x-intercept is when y = 0 ==> x = 0
3x
(b) y = -----
x - 3
x - 3 = 0 ==> x = 3 (vertical asymtote)
3x/x 3
y = --------- = ------- ==> y = 3 (horizontal asymtote)
x/x - 3/x 1 - 3/x
y-intercept is when x = 0 ==> y = 0
x-intercept is when y = 0 ==> x = 0
2x - 4
(c) y = ------
x
x = 0 (vertical asymtote)
2x/x - 4/x 2 - 4/x
y = ---------- = ------- ==> y = 2 (horizontal asymtote)
x/x 1
y-intercept is when x = 0 but this is a vertical asymtote. Therefore no y-intercept.
x-intercept is when y = 0 ==> 2x - 4 = 0 ==> x = 2
1 - x
(d) y = -----
x
x = 0 (vertical asymtote)
1/x - x/x 1/x - 1
y = --------- = ------- ==> y = -1 (horizontal asymtote)
x/x 1
y-intercept is when x = 0 but this is a vertical asymtote. Therefore no y-intercept.
x-intercept is when y = 0 ==> 1 - x = 0 ==> x = 1
