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
x4
3x
f x
x4
y3
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
y0
4x 3
f x 3
x 1
4x 3
f x 3
x 1
y4
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
x2
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)