1. The Parent Function can be transformed by using
What do a, h and k represent?
a the vertical stretch or compression
h the horizontal translation
k the vertical translation

2. There can be vertical asymptotes horizontal asymptotes
What is an asymptote?
The line where a graph approaches
There can be vertical asymptotes horizontal asymptotes
and
slant asymptotes.
What is the vertical asymptote?
What is the horizontal asymptote?

3. Example 1 What is the transformation? What is the vertical asymptote?
Down 3 units
What is the vertical asymptote?
What is the horizontal asymptote?

4. Example 1
What is the Domain?
What is the Range?

5. Example 2 What is the transformation? What is the vertical asymptote?
Left 2 units
What is the vertical asymptote?
What is the horizontal asymptote?

6. To Graph Rational Functions
Factor Completely
Vertical Asymptotes: set the denominator
equal to zero. It is stated as an equation
The vertical asymptotes are the domain
exclusions.
3. Horizontal Asymptotes: compare the degrees of the numerator and denominator.

7. Horizontal Asymptote If then HA: y = 0 If then HA: y = 2 If HA: none
you look at
then
HA: y = 2 If
then there is NO horizontal asymptote
HA: none

8. Graphing continued What is another name for the x-intercepts?
4. Roots: set the numerator = 0
What is another name for the x-intercepts?
Roots
What does it mean on the graph?
Where the graph crosses the x-axis

9. Graphing continued 5. Holes:
Look at factors that are common to numerator
and denominator. Set those equal to 0 and solve.
Vertical Asymptotes do NOT include the holes.

10. 8-4: Graphs of Rational Functions [Day 1]
Example 3
Example 2
Identify the following for
Vertical asymptote
Horizontal Asymptote
Roots:
Holes:
Domain:
Range:
Y-intercept:
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11. 8-4: Graphs of Rational Functions [Day 1]
Example 4
Identify the following for
Vertical asymptote
Horizontal Asymptote
Roots:
Holes:
Y-intercept:
Vertical asymptote
Roots
Horizontal Asymptote
look at the degrees
Same Degree
Divide the leading
coefficients.
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12. 8-4: Graphs of Rational Functions [Day 1]
Example 4
graph
Zeros:
Vertical Asymptote:
Horizontal Asymptote:
Domain:
Range:
x = –2
y = 1
8-4: Graphs of Rational Functions [Day 1] 12

13. 8-4: Graphs of Rational Functions [Day 1]
Example 5
Identify the following for
Vertical asymptote
Roots
Vertical asymptote
Horizontal Asymptote
Roots:
Holes:
Y-intercept:
Horizontal Asymptote
Holes
look at the degrees
8-4: Graphs of Rational Functions [Day 1]
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14. 8.4 - Graphs of Rational Functions [Day 2]
Example 5
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15. 8-4: Graphs of Rational Functions [Day 1]
Example 6
Identify the following for
Vertical asymptote
Roots
Vertical asymptote
Horizontal Asymptote
Roots:
Holes:
Horizontal Asymptote
Holes
look at the degrees
8-4: Graphs of Rational Functions [Day 1]
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16. 8.4 - Graphs of Rational Functions [Day 2]
Example 6
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17. Extra problems following this slide

18. 8.4 - Graphs of Rational Functions [Day 2]
Example 7
Given, graph, determine the zeros, all asymptotes/holes, and domain
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19. 8.4 - Graphs of Rational Functions [Day 2]
Example 8
Given, graph, determine the zeros and all asymptotes/holes
Hole at x = 3
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20. 8.4 - Graphs of Rational Functions [Day 2]
Without a Calculator
Identify the zeros and determine all of the asymptotes to the following:
4x – 12
x – 1
1. f(x) =
(x + 2)(x – 3)
x + 1
2. f(x) =
x – 2
x2 + x
3. f(x) =
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21. 8-4: Graphs of Rational Functions [Day 1]
Example 3
Your Turn
Your Turn
Identify the zeros and vertical asymptote(s) of
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22. Horizontal Asymptotes: compare the degrees of the numerator and denominator.
Degree of Numerator > Degree of Denominator
Top bigger---NO…means NO horizontal asymptotes. Must find Oblique asymptote.
Degree of Denominator > Degree of Numerator
Bottom bigger---OH…means horizontal asymptote is y=0.
Degree of Numerator = Degree of Denominator
-CO…means find ratio of leading coefficients

23. 8-4: Graphs of Rational Functions [Day 1]
Example 5
Identify the following for
Vertical asymptote
Roots
Vertical asymptote
Horizontal Asymptote
Roots:
Holes:
Horizontal Asymptote
look at the degrees
Same Degree
Divide the leading
coefficients.
8-4: Graphs of Rational Functions [Day 1]
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24. 8-4: Graphs of Rational Functions [Day 1]
Example 5
Given
Zeros:
Vertical Asymptote:
Horizontal Asymptote:
Domain:
Range:
x = –2
x = 2
y = 2
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