Graphs of Rational Functions

1
Graphs of Rational Functions

• Many Rational Functions have graphs which are
  Hyperbolas.
• The most basic Rational Function is y 1/x.
• Most Rational Functions have
  a vertical asymptote.
• A major application of Rational Functions is for
  Inverse Variations/Proportions.

2
Graphs of Rational Functions

• Rational Functions are UNDEFINED at values of x
  which make the denominator 0.
• Recall
  Zero on top OKAY
  Zero on bottom NO WAY
• Non-zero/Zero Vertical Asymptote
• Zero/Zero Hole

3
Vertical Asymptotes vs. Holes

• Ex.
• Ex.

4
Lesson 3 Contents
Example 1 Vertical Asymptotes and Point
Discontinuity Example 2 Graph with a Vertical
Asymptote Example 3 Graph with Point
Discontinuity Example 4 Use Graphs of Rational
Functions
5
Example 3-1a
First factor the numerator and denominator of the
rational expression.
6
Example 3-1b
7
Example 3-1c
Answer vertical asymptote x 5 hole x 3
8
Example 3-2a
9
Example 3-2b
Make a table of values.
Answer
Plot the points and draw the graph.
10
Example 3-2c
As x increases, it appears that the y values of
the function get closer and closer to 1. The line
with the equation f (x) 1 is a horizontal
asymptote of the function.
Answer
11
Example 3-2d
12
Example 3-3a
13
Example 3-3b
14
Example 3-3c
15
Example 3-4a
16
Example 3-4b
17
Example 3-4c
What is the V-intercept of the graph?
Answer The V-intercept is 30.
18
Example 3-4d
What values of t1 and V are meaningful in the
context of the problem?
Answer In the problem context, time and velocity
are positive values. Therefore, positive values
of t1 and V values between 30 and 60 are
meaningful.
19
Example 3-4e
20
Example 3-4f
21
Example 3-4g
Answer The V-intercept is 30.
Answer t1 is positive and V is between 30 and 60.