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Graphs of Rational Functions
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LINEAR FUNCTIONS: SLOPE, GRAPHS AND MODELS
Solving Inequalities with Absolute Values
Solving Inequalities
Solving Equations & Inequalities
Graph the rational function
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Inequalities
Using MATLAB to Solve Linear Inequalities
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Graph Linear Inequalities in Two Variables
Solving Equations & Inequalities
Teaching Inequalities:A Hypothetical Classroom Case
Graphing Linear Inequalities and Systems of Inequalities
Inequalities and Applications
Solving Inequalities
Quadratic Inequalities
Inequalities
Solving Systems of Linear Equations by Graphing
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Inequalities
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Graphing Linear Inequalities
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Solving Inequalities

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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.