1. Rational
Functions
Section 8.3

2. 8.3: Graphing Rational Functions Day 1
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3. 8.3: Graphing Rational Functions Day 1
Intercepts
If f has a y–intercept, it occurs at f(0). In other words, to get the y–intercept, substitute x with zero. Write the y–intercept as a point.
The zeros of the graph of the rational functions occur at:
The zeros of the numerator
The non–zeros of the denominator
Continuity is breaks in the graph of a rational function where the function is NOT defined with the exception for the breaks of the graph. The function is a continuous unbroken curve. Also, the graph has no sharp corners.
8.3: Graphing Rational Functions Day 1
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4. 8.3: Graphing Rational Functions Day 1
Example 1
Identify the y–intercept and the zeros of
Y–intercept: Plug where x = 0
8.3: Graphing Rational Functions Day 1
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5. 8.3: Graphing Rational Functions Day 1
Example 1
Identify the y–intercept and the zeros of
Zeros: Factor and solve for each equation to zero
8.3: Graphing Rational Functions Day 1
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6. 8.3: Graphing Rational Functions Day 1
Example 2
Identify the y–intercept and the zeros of
8.3: Graphing Rational Functions Day 1
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7. 8.3: Graphing Rational Functions Day 1
Your Turn
Identify the y–intercept and the zeros of
8.3: Graphing Rational Functions Day 1
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8. 8.3: Graphing Rational Functions Day 1
Example 3
Identify the domain of the function, and write it in Interval Notation
8.3: Graphing Rational Functions Day 1
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9. 8.3: Graphing Rational Functions Day 1
Simplify
NO!
DO NOT do it this way!
8.3: Graphing Rational Functions Day 1
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10. 8.3: Graphing Rational Functions Day 1
Example 4
Identify the domain of the function, and write it in Interval Notation
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8.3: Graphing Rational Functions Day 1

11. 8.3: Graphing Rational Functions Day 1
Your Turn
Identify the domain of the function, and write it in Interval Notation
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8.3: Graphing Rational Functions Day 1

12. 8.3: Graphing Rational Functions Day 1
Vertical Asymptotes
Vertical Asymptotes are found by making the denominator equal to zero. It is what the restrictions are.
A rational function has a vertical asymptote at x = c provided that c:
C is a zero of the denominator
C is not a zero of the numerator
Graph:
Putting the numerator equal to zero to determine its roots/zeros
Making the denominator equal to zero to find the vertical asymptotes
Follow the rules for horizontal asymptotes
Drawing a dotted line onto the graph to represent the vertical and horizontal asymptote
8.3: Graphing Rational Functions Day 1
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13. 8.3: Graphing Rational Functions Day 1
Example 5
Identify the vertical asymptotes of
Read the question. It is not asking for the domain. It is asking for the vertical asymptote.
8.3: Graphing Rational Functions Day 1
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14. 8.3: Graphing Rational Functions Day 1
Example 5
Identify the vertical asymptotes of
8.3: Graphing Rational Functions Day 1
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15. 8.3: Graphing Rational Functions Day 1
Your Turn
Identify the vertical asymptotes of
8.3: Graphing Rational Functions Day 1
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16. Horizontal Asymptotes
Horizontal Asymptotes are found by looking at the degrees of p(x) and q(x).
Based on the degrees of top and bottom:
Bottom is Bigger: There is a horizontal asymptote at ZERO.
Bottom = Top: Divide the Leading Coefficients
Bottom is Smaller: There is NOT a horizontal asymptote but there COULD be a SLANT Asymptote. To find the slant asymptote, apply SYNTHETIC or LONG DIVISION; Drop of the remainder and keep the equation
The function can NEVER cross a vertical asymptote but it COULD cross a horizontal asymptote once then it will approach the asymptote from the other side as you move towards infinity.
8.3: Graphing Rational Functions Day 1
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17. 8.3: Graphing Rational Functions Day 1
Example 6
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes.
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8.3: Graphing Rational Functions Day 1

18. 8.3: Graphing Rational Functions Day 1
Example 6
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes.
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8.3: Graphing Rational Functions Day 1

19. 8.3: Graphing Rational Functions Day 1
Example 6
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes.
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8.3: Graphing Rational Functions Day 1

20. 8.3: Graphing Rational Functions Day 1
Example 6
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes.
Same Highest Degree
Divide the leading coefficients.
8.3: Graphing Rational Functions Day 1
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21. 8.3: Graphing Rational Functions Day 1
Example 6
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes.
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8.3: Graphing Rational Functions Day 1

22. 8.3: Graphing Rational Functions Day 1
Example 6
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes. x y –6 2 –4 3 –3 5 –1 6 1 x y –6 –4 –3 –1 2 6
8.3: Graphing Rational Functions Day 1
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23. 8.3: Graphing Rational Functions Day 1
Example 7
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes.
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8.3: Graphing Rational Functions Day 1

24. 8.3: Graphing Rational Functions Day 1
Example 7
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes
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8.3: Graphing Rational Functions Day 1

25. 8.3: Graphing Rational Functions Day 1
Example 7
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes
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8.3: Graphing Rational Functions Day 1

26. 8.3: Graphing Rational Functions Day 1
Example 7
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes
Top Degree Bigger
Slant Asymptote using
Synthetic Division
8.3: Graphing Rational Functions Day 1
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27. 8.3: Graphing Rational Functions Day 1
Example 7
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes
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8.3: Graphing Rational Functions Day 1

28. 8.3: Graphing Rational Functions Day 1
Example 7
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes
8.3: Graphing Rational Functions Day 1
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29. 8.3: Graphing Rational Functions Day 1
Example 7
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes x y
–10 –6 –4 –1 –2 1 3 20 x y
–10 –4 –2 –1 1 3
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8.3: Graphing Rational Functions Day 1

30. 8.3: Graphing Rational Functions Day 1
Example 8
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes.
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8.3: Graphing Rational Functions Day 1

31. 8.3: Graphing Rational Functions Day 1
Example 8
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes
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8.3: Graphing Rational Functions Day 1

32. 8.3: Graphing Rational Functions Day 1
Example 8
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes
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8.3: Graphing Rational Functions Day 1

33. 8.3: Graphing Rational Functions Day 1
Example 8
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes
Bottom Degree Bigger
Y = 0
8.3: Graphing Rational Functions Day 1
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34. 8.3: Graphing Rational Functions Day 1
Example 8
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes
8.3: Graphing Rational Functions Day 1
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35. 8.3: Graphing Rational Functions Day 1
Example 8
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes x y –5
–2/21 –4
–1/12 –3 –2
Und. –1
–2/3
–3/4 1
–1 1/3 2 3
1 1/5 x y –5 –4 –3 –2 –1 1 2 3
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8.3: Graphing Rational Functions Day 1

36. 8.3: Graphing Rational Functions Day 1
Your Turn
Graph, Identify the zeros, y–intercept, and vertical and horizontal asymptotes
Zeros:
Y–intercept:
Vertical Asymptote:
Horizontal Asymptote:
8.3: Graphing Rational Functions Day 1
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37. 8.3: Graphing Rational Functions Day 1
Assignment
WKST 1
8.3: Graphing Rational Functions Day 1
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