Discontinuities
1.3
2006/03/02 22:45:05 US/Central
2006/05/01 18:49:56.886 GMT-5
Pradnya
Vinay
Bhawalkar
pradnyab89@gmail.com
Kim
Dodge
Johnston
kim.johnston@fortbend.k12.tx.us
Pradnya
Vinay
Bhawalkar
pradnyab89@gmail.com
Asymptote
Discontinuity
Holes
Rational function
Vertical Asymptote
Finding discontinuities - vertical asymptotes and holes - of rational functions
Vertical Asymptotes occur when factors in the denominator = 0 and do not cancel with factors in the numerator
- Vertical asymptotes are vertical lines the graph approaches
- The equation of the vertical asymptote is x = (that number which makes the denominator = 0)
Holes (Removable Discontinuities) occur when the factor in the denominator = 0 and it cancels with like factors in the numerator.
- Holes are open "points" so they have an x and y coordinate
- The x-value is the number that makes the cancelled factor = 0.
- The y-value is found by substituting x into the "reduced" equation (after cancelling) like factors.
Find the vertical asymptotes and holes (if any) for the following. Don't forget that vertical asymptotes are equations and holes are points!y1x
Vertical Asymptote: x0Hole: None
y
x
x
1
x
1
Vertical Asymptote: NoneHole: (1,1) since (x-1) was cancelled, the hole is at x=1. To find the y-coordinate, plug 1 into the reduced equation:
x
x
1
x
1
x
1
y
4
x
3
x
7
Vertical Asymptote: x7
since
x70
Hole: None
y9x32x
Vertical Asymptote: x32
since 32x0, x32
Hole: None
y
7
x
9
x
1
Vertical Asymptote:
x9
,
x1
since
x
9
and
x
1
Hole: None
y
7
x
2
x
2
7
x
3
Vertical Asymptote:
x
12,
x3
since
2
x
2
7
x
3
0
,
2x
1
x
3
0
,
2x
1
0
and
x
3
0
,
x
12
and
x
3
Hole: None
y
2
x
1
x
5
2
-1
Vertical Asymptote:
x
5
since
x5
2
0
,
x5
0
,
x
5
Hole: None
y
x
3
x
2
25
-1
Vertical Asymptote: None since
x
2
25
0
,
x2
25
, a number squared will never be negative
Hole: None
y
x
7
x
2
2
-1
Vertical Asymptote: None since
x
2
2
0
,
x2
2
and any number squared will never be a negative number
Hole: None
y
5
x3
Vertical Asymptote: x3
since
x3
0
,
x3
0
,
x
3
Hole: None
y
4
x
4
Vertical asymptotes:
x
-4
and
x
4
since
x
4
0
,
x
4
,
x
-4
and
x
4
Hole: None
y
3
x
2
x
6
4
x
2
9
Vertical Asymptote: x-3
Hole: (3, 58)
since
3
x
2
x
6
4
x
2
9
3
x
3
x
2
4
x
3
x
3
3
x
2
4
x
3
,
(x-3) was cancelled, so the hole is at x=3. To find the y-coordinate, plug 3 into the reduced equation:
3
3
2
4
3
3
3
5
4
6
15
24
5
8
y
-2
x
2
4
3
x
2
4
x
4
-2
x
2
4
3
x
2
4
x
4
-2
x
2
x
2
3
x
2
2
-2
x
2
3
x
2
Vertical Asymptote:
x-2
Hole: None
since the vertical asymptote takes care of the hole.
y
x
2
4
x
2
Vertical Asymptote: None
Hole: (-2,-4) since
x
2
4
x
2
x
2
x
2
x
2
x
2
, (x+2) was cancelled, so the hole is at x = -2. To find the y-coordinate, plug -2 into the reduced equation:
-2
2
-4
y
x
2
x
3
x
2
3
x
Vertical Asymptotes: None
Holes: (3,3), (0,0) since
x
2
x
3
x
2
3
x
x
2
x
3
x
x
3
x
, x and (x-3) were cancelled, so the holes are at x=0 and x=3. To find the y-coordinate, plug 0 and 3 into the reduced equation: 0, 3
y
x
3
1
x
1
Vertical Asymptote: None
Hole: (1,3) since
x
3
1
x
1
x
1
x
2
x
1
x
1
x
2
x
1
, (x-1) was cancelled, so the hole is at x=1. To find the y-coordinate, plug 1 into the reduced equation:
1
2
1
1
3
y
2
x
2
3
x
5
x
2
1
2
x
2
3
x
5
x
2
1
2
x
5
x
1
x
1
x
1
2
x
5
x
1
Vertical asymptote:
x
1
since
x
1
0
Hole: (-1,
7
2
)
Since (x+1) was cancelled, the hole is at x= -1. To find the y-coordinate, plug -1 into the reduced equation:
2
-1
5
-1
1
72