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
T: (togetherness about the asymptotes)
     ______                                                                                                                             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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