In this section we will expand our knowledge on the connection between derivatives and the shape of a graph. By following the "5-Steps Approach", we will quantify the characteristics of the function with application of derivatives, which will enable us to sketch the graph of a function.

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.

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.