1. Rational Functions and Asymptotes
Arrow notation:
Meaning:
x approaches a from the right
x approaches a from the left
x is approaching infinity, increasing forever
x is approaching infinity, decreasing forever ,

2. Vertical Asymptotes
The line x = a is a vertical asymptote of a function if
f(x) increases or decreases without bound as x approaches a.
If as
then x = a is vertical asymptote of the function.
Horizontal Asymptotes
The line y = b is a horizontal asymptote of a function if f(x) approaches b as x increases or decreases without bound.
If as

3. Vertical asymptotes are a result of a domain issue.
Find these by setting the denominator equal to zero.
Horizontal asymptotes guide the end behavior of graph.
If the degree of the numerator < degree of the denominator then the horizontal asymptote will be y = 0.
If the degree of the numerator = degree of denominator then the horizontal asymptote will be
If the degree of the numerator > degree of the denominator
then there is no horizontal asymptote.

4. Graphing a rational function.
Find the vertical asymptotes if they exist.
Find the horizontal asymptote if it exists.
Find the x intercept(s) by finding f(x) = 0.
Find the y intercept by finding f(0).
Find any crossing points on the horizontal asymptote by setting f(x) = the horizontal asymptote.
Graph the asymptotes, intercepts, and crossing points.
Find additional points in each section of the graph as needed.
Make a smooth curve in each section through known points and following asymptotes and end behavior.
Homework: Practice # 1
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