1. Rational
Functions
Section 8.3
Day 2

2. Holes Holes are omitted points in a graph.
In some cases, both the numerator and the denominator of a rational function will equal 0 for a particular value of x.
A function is continuous with the exception for breaks.
The difference between an asymptote and hole is:
Holes are applied to cancelled expressions which cause the graph to be discontinuous
For asymptote graphs, they generally approach the asymptote
A hole CANNOT be a vertical asymptote
A hole CANNOT be a zero
HOLES TAKE PRECEDENCE OVER ASYMPTOTES
8.3: Graphing Rational Functions Day 2
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3. 8.3: Graphing Rational Functions Day 2
Example 1
Identify any holes and vertical asymptotes of
To identify any holes, factor the function out.
8.3: Graphing Rational Functions Day 2
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4. 8.3: Graphing Rational Functions Day 2
Example 1
Identify any holes and vertical asymptotes of
8.3: Graphing Rational Functions Day 2
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5. 8.3: Graphing Rational Functions Day 2
Example 1
Identify any holes and vertical asymptotes of
What is leftover on bottom is the vertical asymptote
8.3: Graphing Rational Functions Day 2
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6. 8.3: Graphing Rational Functions Day 2
Example 2
Graph, Identify the zeros, y–intercept, vertical and horizontal asymptotes, and holes.
Hole(s):
Zero(s):
Y–intercept:
Vertical Asymptote:
Horizontal Asymptote: x y –4
1/3 –3 –2 –1
Und 3 1 2
1.667
3/2 4
1.4 x y –4 –3 –2 –1 1 2 3 4
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8.3: Graphing Rational Functions Day 2

7. 8.3: Graphing Rational Functions Day 2
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8. 8.3: Graphing Rational Functions Day 2
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9. 8.3: Graphing Rational Functions Day 2
Example 3
Graph, Identify the zeros, y–intercept, vertical and horizontal asymptotes, and holes. x y –3 –2 –1 1 2 3 4 5 x y –3
–1.6 –2
–.75 –1 1 2
Und 3 8 4
7.5 5
Hole(s):
Zero(s):
Y–intercept:
Vertical Asymptote:
Horizontal Asymptote:
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8.3: Graphing Rational Functions Day 2

10. 8.3: Graphing Rational Functions Day 2
Your Turn
Graph, Identify the zeros, y–intercept, vertical and horizontal asymptotes, and holes.
Hole(s):
Zero(s):
Y–intercept:
Vertical Asymptote:
Horizontal Asymptote: x y –3
–0.6 –2
–0. 5 –1
1/3 1 2
Und 3 4 x y –3 –2 –1 1 2 3 4
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8.3: Graphing Rational Functions Day 2

11. 8.3: Graphing Rational Functions Day 2
Example 3
Write an equation where there is a hole at (2, 5/3), zero is at (–3, 0), vertical asymptote is at x = –1, and horizontal asymptote is at y = 1.
8.3: Graphing Rational Functions Day 2
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12. 8.3: Graphing Rational Functions Day 2
Example 4
Write an equation where there is a hole at (–2/3, –7/13), zero is at (1/2, 0), vertical asymptote is at x = –5, and horizontal asymptote is at y = 2.
8.3: Graphing Rational Functions Day 2
5/4/ :17 PM

13. 8.3: Graphing Rational Functions Day 2
Example 5
Write an equation where there is a hole at (1, 1), zero is at (0, 0), vertical asymptote is at x = –1, and horizontal asymptote is at y = 2.
8.3: Graphing Rational Functions Day 2
5/4/ :17 PM

14. 8.3: Graphing Rational Functions Day 2
Your Turn
Write a rational equation where the zero is at (-5, 0), a hole exists at (2, 14/11), vertical asymptote is at x = –7/2, horizontal asymptote is at y = 1, and the y-intercept is at (0, 10/7).
8.3: Graphing Rational Functions Day 2
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15. 8.3: Graphing Rational Functions Day 2
Assignment
WKST 2 (double homework)
8.3: Graphing Rational Functions Day 2
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