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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

Exercise 1

y=4x+3x-7 y 4 x 3 x 7

Solution 1

Vertical Asymptote: x=7x7 since x-7=0 x70

Hole: None

Exercise 2

y=9x3-2x y9x32x

Solution 2

Vertical Asymptote: x=32x32 since 3-2x=032x0, x=32x32

Hole: None

Exercise 3

y=7x-9x+1 y 7 x 9 x 1

Solution 3

Vertical Asymptote: x=9 x9, x=-1 x1 since x=9 x 9 and x=-1 x 1

Hole: None

Exercise 4

y=7x2x2-7x+3 y 7 x 2 x 2 7 x 3

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

Exercise 5

y=2x+1x+52 y 2 x 1 x 5 2 -1

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

Exercise 6

y=x+3x2+25 y x 3 x 2 25 -1

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

Exercise 7

y=x-7x2+2 y x 7 x 2 2 -1

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

Exercise 8

y=5|x-3| y 5 x3

Solution 8

Vertical Asymptote: x=3x3 since |x-3|=0 x3 0 , x-3=0 x3 0 , x=3 x 3

Hole: None

Exercise 9

y=4|x|-4 y 4 x 4

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

Exercise 10

y=3x2-x-64x2-9 y 3 x 2 x 6 4 x 2 9

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

Exercise 11

y=-2x2-43x2+4x+4 y -2 x 2 4 3 x 2 4 x 4

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.

Exercise 12

y=x2-4x+2 y x 2 4 x 2

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

Exercise 13

y=x2x-3x2-3x y x 2 x 3 x 2 3 x

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

Exercise 14

y=x3-1x-1 y x 3 1 x 1

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

Exercise 15

y=2x2-3x-5x2-1 y 2 x 2 3 x 5 x 2 1

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

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