jeff bivin -- lzhs graphing rational functions jeffrey bivin lake zurich high school...
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Jeff Bivin -- LZHS
Graphing Rational Functions
Jeffrey Bivin
Lake Zurich High School
Last Updated: February 18, 2008
Jeff Bivin -- LZHS
xy
1Graph:
x y
1 1
2
5
10
2
10
101
51
21
21
101
x = 0
y =
0
Jeff Bivin -- LZHS
xy
1Graph:
x y
-1 -1
-2
-5
-10
-2
-10
1015121
21
101
x = 0
y =
0
Jeff Bivin -- LZHS
xy
1Graph:
x = 0
y =
0
DOMAIN
RANGE
,00,
,00,
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph:
x = 2
y =
3
32
1
x
y
x – 2 = 0
x = 2
2
3
Jeff Bivin -- LZHS
Graph:
x = 2
y =
3
y-intercept
32
1
x
y
32
1
x
y0
32
1
y
2
6
2
1
y
2
5y
2
5,0
Jeff Bivin -- LZHS
Graph:
x = 2
y =
3
x-intercept
32
1
x
y
32
1
x
y0
2
13
x
1)2(3 x163 x
0,3
5
53 x
3
5x
1
3
Jeff Bivin -- LZHS
Graph:
x = 2
y =
3
32
1
x
y
DOMAIN
RANGE
,22,
,33,
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph:
x = -1
y =
-2
21
3
x
y
x + 1 = 0
x = -1
1
2
Vertical stretch
Jeff Bivin -- LZHS
Graph:
x = -1
y =
-2
21
3
x
y
y-intercept
21
3
x
y
21
3y
23 y
1y 1,0
0
Jeff Bivin -- LZHS
Graph:
x = -1
y =
-2
21
3
x
y
x-intercept
21
3
x
y0
1
32
x
3)1(2 x322 x
0,2
112 x
2
1x
Jeff Bivin -- LZHS
Graph:
x = -1
y =
-2
21
3
x
y
DOMAIN
RANGE
,11,
,22,
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph:
x = 1
y =
2
1
32
x
xy
x - 1 = 0
x = 1
Horizontal asymptote
Vertical asymptote
1
1
2y
2y
Jeff Bivin -- LZHS
Graph:
x = 1
y =
2
1
32
x
xy
y-intercept
1
32
x
xy
1
3
y
3y 3,0
0
0
Jeff Bivin -- LZHS
Graph:
x = 1
y =
2
1
32
x
xy
x-intercept
1
32
x
xy
320 x
x23
x2
3
0,2
3
01
0
Jeff Bivin -- LZHS
Graph:
x = 1
y =
2
1
32
x
xy
DOMAIN
RANGE
,11,
,22,
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph: 32
15
x
xy
2x - 3 = 0
Horizontal asymptote
Vertical asymptote
2
5y
2
3x
2x = 3
23x
25y
Jeff Bivin -- LZHS
Graph: 32
15
x
xy
23x
25y
y-intercept
32
15
x
xy
3
1
y
3
1y
31,0
0
0
Jeff Bivin -- LZHS
Graph: 32
15
x
xy
23x
25y
x-intercept
32
15
x
xy
150 x
x51
x5
1
0,5
1
01
0
31,0
Jeff Bivin -- LZHS
Graph: 32
15
x
xy
23x
25y
DOMAIN
RANGE
,, 23
23
,, 25
25
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph: 6
12
xx
xy
x - 2 = 0
Horizontal asymptote
Vertical asymptote
0y
x = 2
20x)3)(2( xx
y =
0
x + 3 = 0
x = -3
x = 2x = -3
Jeff Bivin -- LZHS
Graph: 6
12
xx
xy
y =
0
x = 2x = -3
y-intercept
6
1
y
6
1y
61,0
6
12
xx
xy
0
0 0
Jeff Bivin -- LZHS
Graph: 6
12
xx
xy
y =
0
x = 2x = -3
x-intercept
10 x
x 1
0,1
6
12
xx
xy1
0
Jeff Bivin -- LZHS
Graph: 6
12
xx
xy
y =
0
x = 2x = -3
72,5
)3)(2( xx
)35)(25(
15
y
)2)(7(
4
y
14
4y
7
2y
let x = -5
Jeff Bivin -- LZHS
Graph: 6
12
xx
xy
y =
0
x = 2x = -3
233,
)3)(2( xx
)33)(23(
13
y
)6)(1(
4y
6
4y
3
2y
let x = 3
Jeff Bivin -- LZHS
Graph: 6
12
xx
xy
y =
0
x = 2x = -3
)3)(2( xx
DOMAIN
RANGE
,22,33,
,
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph: 4
32
x
y
x - 2 = 0
Horizontal asymptote
Vertical asymptote
0y
x = 2
20x
)2)(2( xx
y =
0
x + 2 = 0
x = -2
x = 2x = -2
Jeff Bivin -- LZHS
22
3
xxy
Graph: 22
3
xxy
y =
0
x = 2
y-intercept
4
3
y
4
3y
43,0
0 0
x = -2
Jeff Bivin -- LZHS
Graph:
y =
0
x = 2x = -2
22
3
xxy
x-intercept
30
22
3
xxy1
0
Ø
Jeff Bivin -- LZHS
Graph:
y =
0
x = 2x = -2
22
3
xxy
53,3
)23)(23(
3
y
)1)(5(
3
y
5
3y
let x = -3
Jeff Bivin -- LZHS
Graph:
y =
0
x = 2x = -2
22
3
xxy
53,3)23)(23(
3
y
)5)(1(
3y
5
3y
let x = 3
Jeff Bivin -- LZHS
Graph:
y =
0
x = 2x = -2
22
3
xxy
DOMAIN
RANGE
,22,22,
,00,
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph: 43
32
xx
xy
x + 4 = 0
Horizontal asymptote
Vertical asymptote
0y
x = -4
20x)1)(4( xx
y =
0
x - 1 = 0
x = 1
x = 1x = -4
Jeff Bivin -- LZHS
Graph: 14
3
xx
xy
y =
0
x = 1x = -4
y-intercept
4
3
y
4
3y
14
3
xx
xy
43,0
1040
30
y
Jeff Bivin -- LZHS
Graph: 14
3
xx
xy
y =
0
x = 1x = -4
x-intercept
14
3
xx
xy
43,0
1
0
30 x
x3
0,3
Jeff Bivin -- LZHS
Graph: 14
3
xx
xy
y =
0
x = 1x = -4
1545
35
y
435,
8
( 1)( 6)y
8 4
6 3y
let x = -5
Jeff Bivin -- LZHS
Graph: 14
3
xx
xy
y =
0
x = 1x = -4
1242
32
y
61,2
)1)(6(
1y
6
1y
let x = 2
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph:
4113
531
xxxx
xxxy
x - 1 = 0
Horizontal asymptote
Vertical asymptote
0y
x = 1
x + 4 = 0
x = -4
41
5
xx
xy
2
20
x
xxy
x - 3 = 0
Holes
x = 3x + 1 = 0
x = -1
4313
53
y
7
4
72
8y
4111
51
y
3
2
32
4
y
74,3 32,1
Jeff Bivin -- LZHS
Graph:
y =
0
x = 1x = -4
Horizontal asymptote
0y
Vertical asymptote
x = 1 x = -4
41
5
xx
xy
Holes
74,3 32,1
Jeff Bivin -- LZHS
Graph:
y =
0
x = 1x = -4
41
5
xx
xy
y-intercept
415
y
4
5y
45,0
4010
50
y
Jeff Bivin -- LZHS
Graph:
y =
0
x = 1x = -4
41
5
xx
xy
x-intercept
41
5
xx
xy1
0
50 x
x 5
0,5
Jeff Bivin -- LZHS
Graph:
y =
0
x = 1x = -4
41
5
xx
xy
VA
HA
Holes
y-int
x-int
0y
74,3 32,1
0,3 45,0
x = 1
x = -4
Jeff Bivin -- LZHS
Graph:
y =
0
x = 1x = -4
41
5
xx
xy
let x = -∞
y
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph:
6 1 2 2 4
2 1 2 3 6
x x x x xy
x x x x x
x - 1 = 0
Horizontal asymptote
Vertical asymptote
1y
x = 1
x + 3 = 0
x = -3
1 4
1 3
x xy
x x
2
2
x
xy
Jeff Bivin -- LZHS
Graph:
31
41
xx
xxy
x - 2 = 0
Holes
x = 2x + 2 = 0
x = -2 3212
4212
y
5
6
51
23
y
3212
4212
y
2
13
61
y
56,2 2,2
x + 6 = 0x = -6 3616
4616
y
21
50
37
105
y
2150,6
6 1 2 2 4
2 1 2 3 6
x x x x xy
x x x x x
Jeff Bivin -- LZHS
Graph:
1 4
1 3
x xy
x x
y-intercept
31
41
y
3
4y
0 1 0 4
0 1 0 3y
6 1 2 2 4
2 1 2 3 6
x x x x xy
x x x x x
430,
Jeff Bivin -- LZHS
6 1 2 2 4
2 1 2 3 6
x x x x xy
x x x x x
Graph:
1 4
1 3
x xy
x x
x-intercept(s)
31
31
xx
xxy
310 xx
01x
0,11
0
1x03x3x
4, 0
Jeff Bivin -- LZHS
Graph:
y =
1
x = 1x = -3
VA
HA
Holes
y-int
x-int
1y
0,1 34,0
x = 1
x = -3
1 4
1 3
x xy
x x
4, 0
56,2 2,2 21
50,6
Jeff Bivin -- LZHS
6 1 2 2 4
2 1 2 3 6
x x x x xy
x x x x x
Graph:
1 4
1 3
x xy
x x
Asymptote C
ross
ing(s)
4332 22 xxxx
4332 xx
4131 xxxx
1
1
15 x
51x
1,51
Jeff Bivin -- LZHS
Graph:
y =
1
x = 1x = -3
VA
HA
Holes
y-int
x-int
1y
0,1 34,0
x = 1
x = -3
31
31
xx
xxy
0,3
56,2 2,2 21
50,6
AX 1,51
Jeff Bivin -- LZHS
Graph:
y =
1
x = 1x = -3
31
31
xx
xxy
DOMAIN
RANGE
,22,11,22,33,66,
,
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph:
x - 3 = 0
Horizontal asymptote
Vertical asymptote
0y
x = 3
x + 2 = 0
x = -2
23
3
xxy
2
20
x
xy
Jeff Bivin -- LZHS
Graph:
y-intercept
23
3
xxy
233
y
2
1y
2030
3
y
21,0
x-intercept
23
3
1
0
xx
30
Ø
Jeff Bivin -- LZHS
Graph:
y =
0
x = 3x = -2
23
3
xxy
VA
HA
Holes none
y-int
x-int none
0y
x = 3
x = -2
21,0
Jeff Bivin -- LZHS
Graph:
y =
0
x = 3x = -2
23
3
xxy
let x = -∞
y
let x = +∞
y
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph:
Horizontal asymptote
Vertical asymptote
0y
1
42
x
y
2
20
x
xy
012 x
12 x
ix
Jeff Bivin -- LZHS
Graph:
y-intercept
1
4y
4y
10
4
y
4,0
x-intercept
1
4
1
02
x
40
Ø
1
42
x
y
Jeff Bivin -- LZHS
Graph:
y =
0
VA none
HA
Holes none
y-int
x-int none
0y
4,0
1
42
x
y
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph:
1
12
x
xxy
2 xy
Hole
x + 1 = 0
x = -1
y-intercept
y = 0 - 2
y = -2
x-intercept
0 = x - 2
2 = x
3,1
2,0 0,221y3y
Jeff Bivin -- LZHS
Graph:
VA none
HA none
Holes
y-int
x-int
2,0
3,1
0,2
2 xy
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph:
1
42
x
xxy
2
2
0x
xy
Horizontal asymptote: none
1
822
x
xxy
821 2 xxx
1x
xx 2
8x1x
9Slan
t asy
mpt
ote
1xy
Vertical asymptote
x + 1 = 0
x = -1
Jeff Bivin -- LZHS
Graph:
1
42
x
xxy
y-intercept
8,0
10
4020
y
1
42y
8y
x-intercept(s)
1
42
1
0
x
xx
420 xx
02x
0,22x
04x4x
0,4
Jeff Bivin -- LZHS
Graph: y =
x+1x = -1
VA
HA none
SA
Holes none
y-int
x-int
AX
x = -1
y=x+1
8,0
0,2 0,4
1
42
x
xxy
Jeff Bivin -- LZHS
Graph:
1
42
x
xxy
Asymptote crossing
1
421
x
xxx
4211 xxxx
8212 22 xxxx
81
Ø
Jeff Bivin -- LZHS
Graph: y =
x+1x = -1
VA
HA none
SA
Holes none
y-int
x-int
AX none
x = -1
y=x+1
8,0
0,2 0,4
1
42
x
xxy
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Graph:
)1(1
1
xx
xy
1
1
x
y
x + 1 = 0
Horizontal asymptote
Vertical asymptote
0y
x = -1
x
xy
0
Hole
No holes
x + 1 = 0
x = -1Since both a vertical asymptote and a hole seem to
be at x = - 1, the asymptote wins.
Jeff Bivin -- LZHS
Graph:
)1(1
1
xx
xy
1
1
x
y
y-intercept x-intercept
1,0
10
1
y
1y
1
1
1
0
x
10
none
Jeff Bivin -- LZHS
Graph:
x = -1
VA
HA
SA none
Holes none
y-int
x-int none
AX none
x = -1
1,0
0y
y =
0
1
1
)1(1
1
xxx
xy