Connexions

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Discontinuities

Module by: Pradnya Bhawalkar, Kim Johnston

Summary: 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!
Example 1 
y=1x y1x
Vertical Asymptote: x=0x0
Hole: None
Example 2 
y=xx-1x-1 y x x 1 x 1
Vertical Asymptote: None
Hole: (1,1) since (x-1) was cancelled, the hole is at x=1. To find the y-coordinate, plug 1 into the reduced equation: xx-1x-1=x=1 x x 1 x 1 x 1
Problem 1
y=4x+3x-7 y 4 x 3 x 7
[ Click for Solution 1 ]
Solution 1
Vertical Asymptote: x=7x7 since x-7=0 x70
Hole: None
[ Hide Solution 1 ]
Problem 2
y=9x3-2x y9x32x
[ Click for Solution 2 ]
Solution 2
Vertical Asymptote: x=32x32 since 3-2x=032x0, x=32x32
Hole: None
[ Hide Solution 2 ]
Problem 3
y=7x-9x+1 y 7 x 9 x 1
[ Click for Solution 3 ]
Solution 3
Vertical Asymptote: x=9 x9, x=-1 x1 since x=9 x 9 and x=-1 x 1
Hole: None
[ Hide Solution 3 ]
Problem 4
y=7x2x2-7x+3 y 7 x 2 x 2 7 x 3
[ Click for Solution 4 ]
Solution 4
Vertical Asymptote: x=12 x 12, x=3 x3 since 2x2-7x+3=0 2 x 2 7 x 3 0 , 2x-1x-3=0 2x 1 x 3 0 , 2x-1=0 2x 1 0 and x-3=0 x 3 0 , x=12 x 12 and x=3 x 3
Hole: None
[ Hide Solution 4 ]
Problem 5
y=2x+1x+52 y 2 x 1 x 5 2 -1
[ Click for Solution 5 ]
Solution 5
Vertical Asymptote: x=-5 x 5 since x+52=0 x5 2 0 , x+5=0 x5 0 , x=-5 x 5
Hole: None
[ Hide Solution 5 ]
Problem 6
y=x+3x2+25 y x 3 x 2 25 -1
[ Click for Solution 6 ]
Solution 6
Vertical Asymptote: None since x2+25=0 x 2 25 0 , x2=-25 x2 25 , a number squared will never be negative
Hole: None
[ Hide Solution 6 ]
Problem 7
y=x-7x2+2 y x 7 x 2 2 -1
[ Click for Solution 7 ]
Solution 7
Vertical Asymptote: None since x2+2=0 x 2 2 0 , x2=-2 x2 2 and any number squared will never be a negative number
Hole: None
[ Hide Solution 7 ]
Problem 8
y=5|x-3| y 5 x3
[ Click for Solution 8 ]
Solution 8
Vertical Asymptote: x=3x3 since |x-3|=0 x3 0 , x-3=0 x3 0 , x=3 x 3
Hole: None
[ Hide Solution 8 ]
Problem 9
y=4|x|-4 y 4 x 4
[ Click for Solution 9 ]
Solution 9
Vertical asymptotes: x=-4 x -4 and x=4 x 4 since |x|-4=0 x 4 0 , |x|=4 x 4 , x=-4 x -4 and x=4 x 4
Hole: None
[ Hide Solution 9 ]
Problem 10
y=3x2-x-64x2-9 y 3 x 2 x 6 4 x 2 9
[ Click for Solution 10 ]
Solution 10
Vertical Asymptote: x=-3x-3
Hole: (3, 5858) since 3x2-x-64x2-9=3x-3x+24x+3x-3=3x+24x+3 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: 33+243+3=3×54×6=1524=58 3 3 2 4 3 3 3 5 4 6 15 24 5 8
[ Hide Solution 10 ]
Problem 11
y=-2x2-43x2+4x+4 y -2 x 2 4 3 x 2 4 x 4
[ Click for Solution 11 ]
Solution 11
-2x2-43x2+4x+4=-2x+2x-23x+22=-2x-23x+2 -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=-2x-2
Hole: None since the vertical asymptote takes care of the hole.
[ Hide Solution 11 ]
Problem 12
y=x2-4x+2 y x 2 4 x 2
[ Click for Solution 12 ]
Solution 12
Vertical Asymptote: None
Hole: (-2,-4) since x2-4x+2=x+2x-2x+2=x-2 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 -2 2 -4
[ Hide Solution 12 ]
Problem 13
y=x2x-3x2-3x y x 2 x 3 x 2 3 x
[ Click for Solution 13 ]
Solution 13
Vertical Asymptotes: None
Holes: (3,3), (0,0) since x2x-3x2-3x=x2x-3xx-3=x 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
[ Hide Solution 13 ]
Problem 14
y=x3-1x-1 y x 3 1 x 1
[ Click for Solution 14 ]
Solution 14
Vertical Asymptote: None
Hole: (1,3) since x3-1x-1=x-1x2+x+1x-1=x2+x+1 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: 12+1+1=3 1 2 1 1 3
[ Hide Solution 14 ]
Problem 15
y=2x2-3x-5x2-1 y 2 x 2 3 x 5 x 2 1
[ Click for Solution 15 ]
Solution 15
2x2-3x-5x2-1=2x-5x+1x+1x-1=2x-5x-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 x 1 since x-1=0 x 1 0
Hole: (-1, 72 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 2 -1 5 -1 1 72
[ Hide Solution 15 ]

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