1. What is the end behavior?
Rational functions may have asymptotes. The function f(x) = has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. 1 x
Parent Function
What is the end behavior?
What are the intervals of increasing and decreasing?
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2. The rational function f(x) = can be transformed by using methods similar to those used to transform other types of functions. 1 x
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3. Example 1: Transforming Rational Functions
Using the graph of f(x) = as a guide, describe the transformation and graph each function. 1
x + 2 1 x
– 3
A. g(x) =
B. g(x) =
Because h = –2, translate f 2 units left.
Because k = –3, translate f 3 units down.
New Asymptotes, EB and inc/dec?
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4. Because k = 1, translate f 1 unit up.
Check It Out! Example 1
Using the graph of f(x) = as a guide, describe the transformation and graph each function. 1 x 1
x + 4 1 x
+ 1
a. g(x) =
b. g(x) =
Because k = 1, translate f 1 unit up.
Because h = –4, translate f 4 units left.
New Asymptotes, EB and inc/dec?
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5. The values of h and k affect the locations of the asymptotes, the domain, and the range of rational functions whose graphs are hyperbolas.
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6. Example 2: Determining Properties of Hyperbolas
Identify the asymptotes, domain, and range of the function g(x) = – 2. 1
x + 3 1
x – (–3)
g(x) = – 2
h = –3, k = –2.
Vertical asymptote: x = –3
The value of h is –3.
Domain: {x|x ≠ –3}
Horizontal asymptote: y = –2
The value of k is –2.
Range: {y|y ≠ –2}
Check Graph the function on a graphing calculator. The graph suggests that the function has asymptotes at x = –3 and y = –2.
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7. Identify the asymptotes, domain, and range of the function g(x) = – 5.
Check It Out! Example 2
Identify the asymptotes, domain, and range of the function g(x) = – 5. 1
x – 3 1
x – (3)
g(x) = – 5
h = 3, k = –5.
Vertical asymptote: x = 3
The value of h is 3.
Domain: {x|x ≠ 3}
Horizontal asymptote: y = –5
The value of k is –5.
Range: {y|y ≠ –5}
Check Graph the function on a graphing calculator. The graph suggests that the function has asymptotes at x = 3 and y = –5.
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8. A continuous function is a function whose graph has no gaps or breaks.
A discontinuous function is a function whose graph has one or more gaps or breaks.
A continuous function is a function whose graph has no gaps or breaks.
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9. Set the top=0
Set the bottom=0
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10. Some rational functions have a horizontal asymptote
Some rational functions have a horizontal asymptote. The existence and location of a horizontal asymptote depends on the degrees of the polynomials that make up the rational function.
Note that the graph of a rational function can sometimes cross a horizontal asymptote.
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11. Slide 12 of 25

12. Slant Asymptote
If the degree of the top is one bigger than the degree of the bottom then there is a slant asymptote. You will find it by doing long division until you get y = mx + b

13. Identify the zeros and asymptotes of the function.
x – 2
x2 – 1
f(x) =
x – 2
(x – 1)(x + 1)
f(x) =
Zero: 2
Vertical asymptote: x = 1, x = –1
Horizontal asymptote: y = 0
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14. Identify the zeros and asymptotes of the function.
4x – 12
x – 1
f(x) =
4(x – 3)
x – 1
f(x) =
Zero: 3
Vertical asymptote: x = 1
Horizontal asymptote: y = 4
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15. Identify the zeros and asymptotes, y-int of the function. Then graph.
x2 + 2x – 15
x – 1
f(x) =
(x – 3)(x + 5)
x – 1
f(x) =
Zeros: 3 and –5
Vertical asymptote: x = 1
Horizontal asymptote: none
Slant asymptote: y= x + 3
Y-int: (0, 15)
EB and inc/dec?
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16. Check It Out! Example 4a Continued
Identify the zeros and asymptotes of the function. Then graph.
Graph by using a table of values.
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17. Identify the zeros, y-int and asymptotes of the function. Then graph.
3x2 + x
x2 – 9
f(x) =
x(3x – 1)
(x – 3) (x + 3)
f(x) =
Zeros: 0 and – 1 3
Vertical asymptote: x = –3, x = 3
Horizontal asymptote: y = 3
Y-int: (0, 0)
EB and inc/dec?
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18. Check It Out! Example 4c Continued
Graph by using a table of values.
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19. VA always wins
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20. Example 5: Graphing Rational Functions with Holes
Identify holes in the graph of f(x) =
x2 – 9
x – 3
(x – 3)(x + 3)
x – 3
f(x) =
There is a hole in the graph at x = 3.
To find where the hole would be, plug in 3 into the reduced equation.
f(3) = therefore hole is at (3, 6)
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21. Find all zeros, y-int, holes, asymptotes and domain
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22. Identify the domain, zeros, y-int, asymptotes, and holes in the graph of a rational function. Then graph
f(x) =
2x2 + 2x
x2 – 1
factor:
f(x) =
2x(x + 1)
(x + 1)(x – 1)
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24. zero: 2; asymptotes: x = 0, y = 1; hole at x = 1
Lesson Quiz: Part II
3. Identify the zeros, asymptotes, and holes in the graph of Then graph.
x2 – 3x + 2
x2 – x
f(x) =
zero: 2; asymptotes: x = 0, y = 1; hole at x = 1
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25. Steps include: Find zeros, VA, HA, SA, holes,
HW: P Describe transformations, any asymptotes, D & R, EB, inc/dec and then graph, then do all steps on 23, 25, 27, 31, 33, then on (don’t graph any)
Steps include: Find zeros, VA, HA, SA, holes,
y –int, Domain, EB and inc/dec. Plot on graph then use the table from the calculator
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