# Graphs of Rational Functions

 Summarize asymptotes from last time. Vertical Asymptote: Find zeros of denominator. VA are x = k Horizontal Asymptote: [degree numerator] < [degree denominator] y = 0 [degree numerator] = [degree denominator] y = L (ratio of coefficients of dominant terms) [degree numerator] = [degree denominator] + 1 y = mx + b (find by long division) [degree numerator] > [degree denominator] + 1 no linear asymptotes

## Section 4.3 Graphs of Rational Functions

Now, we will graph rational functions. This process is a bit different from the one in the book ut I think
it is easier to do:

1. Factor, state domain, and THEN reduce.
2. Plot intercepts. For x-intercepts, state whether graph touches (multiplicity is even) or
crosses (multiplicity is odd) x-axis.
3. Draw vertical asymptotes (the zeros of the denominator).
4. Draw horizontal asymptotes:

If deg num < deg denom, y = 0 is a horizontal asymptote
If deg num = deg denom, y = L is a horizontal asymptote
If deg num = deg denom + 1, use long division to find oblique asymptote, y = mx + b
5. Plot where the graph crosses a horizontal or oblique asymptote, if it does. That is, solve
the equation:
Function = asymptote
R(x) = y
6. If needed, plot a few extra points.
7. Connect the dots.

Ex: Graph

1. Factor, state domain, and THEN reduce.

Domain:

Cannot reduce.

2. Plot intercepts. For x-intercepts, state whether graph touches (multiplicity is even) or
crosses (multiplicity is odd) x-axis.
x-intercept:

The factor x + 2 has multiplicity 1, which is odd, so graph crosses here.
y-intercept:

3. Draw vertical asymptotes (the zeros of the denominator).
These occur when the denominator is 0. That is, at x = -3 and x = 3.

4. Draw horizontal asymptotes:
Since deg num < deg denom, HA is y = 0

5. Plot where the graph crosses a horizontal or oblique asymptote, if it does. That is, solve
the equation:

 Solve R(x) = y So, the graph crosses the horizontal asymptote at x = –2.

6. If needed, plot a few extra points.
R(-4) = -0.3
R(4) = 0.9

7. Connect the dots.