Guidelines for Curve Sketching
a) domain
b) Intercepts
y-intercept: set x=0 and evaluate y.

x-intercept: set y=0 and solve for x. (skip this step if the equation is difficult to solve)

c) Asymptotes
vertical asymptotes: for rational functions, vertical asymptotes can be located by equating the denominator to 0 after canceling any common factors.

horizontal asymptotes: evaluate

$lim_{x \to \infty } f(x)$ to determine the right-end behavior;
evaluate

$lim_{x \to -\infty } f(x)$ to determine the left-end behavior.

slant asymptotes: for rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator.

d) Compute$f' (x)$
find the

critical numbers:
• use the First Derivative Test to find:

intervals of increase/decrease and

local extrema.
e) Compute$f'' (x)$ • inflection points occur where the direction of concavity changes.
find possible inflection points by equating the

$f'' (x)$ to 0.

•

Concavity Test:
•

inflection points occur where the direction of concavity changes.