Finding equations for Asymptotes
Defining the asymptotes: horizoNtal
vErtical
Numerator dEnominator
VERTICAL ASYMPTOTES (EASY)
* factor the denominator
­ set each factor equal to zero and solve
­ write your vertical asymptote equation
Ex. 5x ­ 6
x2 ­ 1
1. factor: (x ­ 1) (x + 1)
2. factors = 0: x ­ 1 = 0
x + 1 = 0
3. solve ­ write x = 1
x = ­1
vertical asymptote
To confirm this answer go to Table and make sure that the equation
y value says ERROR where these x­values occur
HORIZONTAL ASYMPTOTES (HARDER)
*Compare the degrees of the numerator and denominator
1.­ if the degrees are the same, then you have a horizontal asymptote
at y = (numerator's leading coefficient)
(denominator's leading coefficient)
2.­ if the numerator's degree is smaller (by any margin), then you have a horizontal asymptote at y = 0 (the x­axis)
3.­ if the numerator's degree is greater (by any margin), then you do not
have a horizontal asymptote
***Way to confirm horizontal asymptotes: How to interpret answer under Y
1 column:
in your calculator 1. 2nd TBLSET *if it's any recognizable number that's the asymptote
*if it's E­# then the asymptote is y = 0
2. Change Indpnt: Ask
*if it's E# then there isn't an asymptote
3. 2nd TABLE
4. type in 1000000 enter
5. Change Indpnt: back to Auto
One final noteworthy tidbit: if you factor the numerator and denominator completely and part of the denominator will cancel out with the numerator then your graph will have a HOLE in it
To confirm this answer go to Table and make sure that the y value says ERROR where this x­values occur
1
­2
4
2
x2 ­ 4
√x2 = x √4 = 2
(x + 2) (x ­ 2)
(x + 2) (x ­ 2)
Hole: x ­ 2 = 0
x = 2
Vertical x + 2 = 0
x = ­2
Write the approximate equation for any vertical or horizontal asymptotes or holes.
Ex a)
Ex b)
Ex c)
x = 2
none
none
x = ­5
y = 0
x = 5
x = ­2 and x = 2
y = 0
none
2
3