Transcript Document

Section 7.2 – Rational Functions and Asymptotes
Graphing and Analyzing Rational Functions
•Domain
•Range
•Vertical Asymptotes (x = )
•Horizontal Asymptotes (y = )
•Holes (x, y)
•Intercepts
•Graph
Find the equations of the horizontal asymptotes of:
3x
f x 
x4
3x
f x 
x4
y3
x 2  2x
f x 
3  4x 2
x 2  2x
f x 
3  4x 2
y
x 4  2x 2  1
f x  2
x  x 1
x 4  2x 2  1
f x  2
x  x 1
none
1
f x 
2x
1
f x 
2x
y0
4x 3
f x  3
x 1
4x 3
f x  3
x 1
y4
3x 4  4
f x  3
x  3x
3x 4  4
f x  3
x  3x
none
1
4
f x 
1
x
Domain:  , 0  ,  0,  
Range:
 , 0 ,  0,  
Vertical Asymptotes: x  0
Horizontal Asymptotes: y  0
Holes: none
Intercepts: none
1
x2

f x 
 x  2  x  2  x  2
Domain:  ,  2 ,  2, 2,  2,  
Range:  , 1  ,  1, 0   0,  
4  4 

Vertical Asymptotes: x  2
Horizontal Asymptotes: y  0
1 


2,
Holes: 

4


 1 
Intercepts:  0, 
 2
 1 
 0, 2 


3x  1
f x  2 
1 x
Domain:  , 1 , 1,  
Range:  ,  1 ,  1,  
Vertical Asymptotes: x  1
Horizontal Asymptotes: y  1
Holes: none
Intercepts:  0, 1
 1, 0 
3x  1
0  2
1 x
3x  1
2 
1 x
2  2x  3x  1
x  1
x 2  5x  6  x  2  x  3 

f x  2
x  2x  3
 x  1 x  3 
Domain:  ,  1 ,  1, 3  ,  3,  
1  1 


,
,  , 1 1,  
Range: 

4 4 

Vertical Asymptotes: x  1
Horizontal Asymptotes: y  1
 1
Holes:  3, 
 4
Intercepts:  0,  2 
 2, 0 
Simplify :
3  x  3
3x  9
3


2
x  9  x  3  x  3  x  3
Extension: The graph contains an hole at x = -3
Note: Cancelled and eliminated
Extension: The graph contains an asymptote at x = 3
Note: not eliminated
3  x  3
3x  9
3
f x  2


x 9
 x  3  x  3  x  3
Domain:  ,  3  ,  3, 3  ,  3,  
1   1 

Range:  ,  ,  , 0   0,  
2  2 

Vertical Asymptotes: x  3
Horizontal Asymptotes: y  0
1 

Holes:  3, 
2

Intercepts:  0,  1
Simplify :
4x 2  8x 4x  x  2 x


12x  24 12  x  2 3
Extension: The graph contains a hole at x = -2
Note: cancelled and eliminated
4x 2  8x 4x  x  2 x

f x 

12x  24 12  x  2 3
Domain:  ,  2 ,  2,  
2   2 

,  , 
Range:  ,

3   3


Vertical Asymptotes: none
Horizontal Asymptotes: none
2 

Holes:  2,
3 

Intercepts:  0, 0 
Graph the rational function which has the following characteristics
Vert Asymp at x = 1, x = -3
Horz Asymp at y = 1
Intercepts (-2, 0), (3, 0), (0, 2)
Passes through (-5, 2)
Graph the rational function which has the following characteristics
Vert Asymp at x = 1, x = -1
Horz Asymp at y = 0
Intercepts (0, 0)
Passes through
(-0.7, 1), (0.7, -1),
(-2, -0.5), (2, 0.5)