Factor the numerator and denominator completely
Identify values that make the denominator zero
Cancel any common factors
Find holes from canceled factors
Find vertical asymptotes from remaining denominator zeros
Find horizontal asymptotes using degrees of numerator and denominator
Find slant asymptotes if the numerator degree is exactly one more than the denominator degree
Find x-intercepts by setting the numerator equal to zero
Find the y-intercept by plugging in x = 0
Determine the domain
Plot asymptotes, holes, and intercepts
Test points in each interval between asymptotes and holes
Sketch the branches of the graph
Check end behavior and behavior near asymptotes
Verify the graph matches all key features