# Notes on Section 3.4

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```					NOTES: LESSON 3.4 Continued: RATIONAL FUNCTIONS AND ASYMPTOTES
DATE____________________
REVIEW
Graph y  ( x  2)3  3 and its inverse           12
y

Is the inverse a function?_________               10
8
6
4
2
x
-5   -4        -2                         2       4    5
-2
-4
-6
-8
-10
-12

y
1                1                                    10
Given f ( x)  graph g ( x)      2
x               x3                                     8

                                                       6

4

2

x
-10   -8   -6   -4   -2                 2   4   6   8   10
                   
-2
                                                       -4

-6

-8

-10

SLANT ASYMPTOTES:
Slant asymptotes may occur when the degree of the numerator of a rational function is one
greater than the degree of the denominator. (Functions will not have both slant and
horizontal asymptotes)

To find the location of a slant asymptote, divide the numerator by the denominator and
ignore the remainder, if there is one.
Notes: Lesson 3.4 Continued Rational Functions and Asymptotes                                                      page 2
Determine the location of all the asymptotes and any holes of the following functions.
Find the x and y intercepts then sketch these curves

EXAMPLE 1:
x2  2 x  1
f ( x)                                                Slant asymptote ______________________
x
y
10
Holes:________________
8

R: (roots) ________________                                                                    6

4
A: (Vertical asymptote(s)): __________
2
T: (tangency at the roots) ____________
x
-10      -8        -6   -4    -2                    2       4        6       8       10
T: (togetherness about the asymptotes)                                                     -2
______
-4

E: (end behavior) _______________                                                          -6
Horizontal asymptotes: _________                                                       -8

Y: y-int = _________                                                                      -10

EXAMPLE 2:
x 2  3x  10
y                                                      Slant asymptote________________________
x2                                                                                      y
12
Holes:________________
10

R: (roots) ________________                                                               8

A: (Vertical asymptote(s)): __________                                                    6

T: (tangency at the roots) ____________                                                   4

2
______                                                                                                                             x
-5      -4         -2                             2            4   5

E: (end behavior)__________________                                                       -2
Horizontal asymptotes: _________
-4
-5
Y: y-int = _________
Notes: Lesson 3.4 Continued Rational Functions and Asymptotes                                            page 3

EXAMPLE 3:
( x  4)( x  2)
Graph f ( x)                                           Slant asymptote __________________________
x( x  3)2 ( x  4)
y
Holes:________________                                                                     10

8
R: (roots) ________________
6
A: (Vertical asymptote(s)): __________                                                      4

T: (tangency at the roots) ____________                                                     2

x
T: (togetherness about the asymptotes)                     -10   -8   -6   -4         -2                 2    4   6   8        10
______                                                                                 -2

-4
E: (end behavior)__________________
-6
Horizontal asymptotes: _________
-8
Y: y-int = _________
-10

EXAMPLE 4:
x2  x  6
Graph y                                             Slant asymptote________________________
( x2  2 x)( x2  8 x  16)

Holes:________________                                                                 y
10

R: (roots) ________________                                                      8

6
A: (Vertical asymptote(s)): __________
4
T: (tangency at the roots) ____________                                          2

T: (togetherness about the asymptotes)                                                                                     x
-10   -8    -6   -4   -2                    2       4   6    8   10
______                                                                      -2

E: (end behavior)__________________                                              -4
Horizontal asymptotes: _________                                             -6

Y: y-int = _________                                                             -8

-10
Notes: Lesson 3.4 Continued Rational Functions and Asymptotes                                     page 4
EXAMPLE 5:
3x 2  x  2
Graph f ( x)  2                                         Slant asymptote:______________________
2 x  5x  3
Holes:________________                                                                    y
10
R: (roots) ________________                                                          8

A: (Vertical asymptote(s)): __________                                               6

4
T: (tangency at the roots) ____________
2

T: (togetherness about the asymptotes)                                                                                  x
______                                            -10      -8   -6   -4   -2             2    4       6   8   10
-2

E: (end behavior)__________________                                                  -4
Horizontal asymptotes: _________
-6

Y: y-int = _________                                                                 -8

-10

EXAMPLE 6:
x2  2 x  3
Graph f ( x)                                           Slant asymptote:__________________________
x2
Holes:________________                                                                    y
10
R: (roots) ________________                                                          8

A: (Vertical asymptote(s)): __________                                               6

4
T: (tangency at the roots) ____________
2

T: (togetherness about the asymptotes)                                                                                  x
______                                            -10      -8   -6   -4   -2             2    4       6   8   10
-2

E: (end behavior)__________________                                                  -4
Horizontal asymptotes: _________
-6

Y: y-int = _________                                                                 -8

-10

```
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