Oblique Asymptote
An oblique asymptote, or slant asymptote, is a line that a function approaches as x goes to -∞ or
∞.
It occurs when the degree of the numerator of a rational function is one more than the degree
of the denominator.
To find the oblique asymptote, divide and ignore the remainder.
• Why can we ignore the remainder?
Examples
f ( x)
6 x3 x 2 19 x 2
3x 2 5 x 1
f ( x)
x2
6 x 12
x 4
Graphing Functions
1.
2.
3.
4.
5.
6.
Find the domain.
Find intercepts and asymptotes.
Find the critical numbers of f(x) and f’(x).
Find intervals of increasing/decreasing and relative extrema.
Find intervals of concave up/down and points of inflection.
Graph.
Examples
a. f(x) = 2x3 + 3x2 – 12x
b.
f ( x)
3x 7
x 2
c.
f ( x)
x
2
6
2x 8
d.
f ( x)
x2
6 x 12
x 4
e.
f ( x)
x2 x 3
f. f(x) = 2sinx + sin2x on [0, 2π]