Algebra 2 Horizontal and Diagonal Asymptotes
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More Asymptotes
Horizontal and Diagonal Asymptotes

 

 If the right side of your equation is a fraction

 where the biggest exponent on top of the fraction is the same power
 as the biggest exponent on the bottom of the fraction,
 the equation has a horizontal (side to side) asymptote.
 
 But this one doesn't happen at Y = 0
 
 To find out where it happens, you throw out everything in the problem
 except for the numbers next to the largest power of X.
 That is, the coefficients.
 
 If you don't see a number there (like in, say X 4
 the number is 1
 
 The fraction that these two coefficients make
 is the place where the horizontal asymptote lives.
 
 Example:

 

 
 To find the asymptote, throw away everything
 except the numbers next to the biggest power of X.
 

 

 
 There is one other way that we can have a horizontal asymptote.
 
 This happens when the biggest exponent on the bottom of the fraction
 is bigger than the biggest exponent on the top of the fraction.
 Like this:

 

 
 When this happens, the horizontal asymptote
 is at Y = 0.
 
 Let's graph this one ...
X Y
-10 .005
-5 .016
-3 .034
-1 -0.25
0 -3
1 0
3 .051
5 .020
10 .005
 

 

 
 
 Another interesting thing happens when the biggest exponent on top
 is one more than the biggest exponent on the bottom.
 Like this ...

 

 
 What we do here, is to get rid of all of the smaller exponent terms.
 Just leave the term with the biggest exponent in the numerator
 and also leave the term with the biggest exponent in the denominator,
 like this ...

 

 
 Now just simplify ...

 

 
 So the asymptote happens at the line Y = 2X
 

 

 
 Let's graph the equation.
 Start with a T table ...
X Y
-10 -19.8
-5 -9.61
-3 -3.17
-1 -1.25
0 0.333
1 1.75
3 5.75
5 9.68
10 19.8
 

 

 
 
 In that last one, the biggest exponent on top
 was one bigger than the biggest exponent on the bottom.
 
 What do you think would happen if the biggest exponent on top
 was 2 bigger than the biggest exponent on the bottom??
 Say, something like this ...

 

 
 If we use the same trick we've been using,
 we would get rid of everything but the biggest exponent term
 in the numerator and the biggest exponent term in the denominator.
 Like this ...

 

 and then we simplify ...

 

 and graph ...

 

 Is this a real asymptote?
 
 It depends on who you ask.
 
 Some people say that asymptotes have to be straight lines.
 But if you graph this complex equation, 
 the graph line keeps getting closer to the line Y = 3X 2
 So it works just like any other asymptote.
 
 I say:
 If it walks like an asymptote,
 and quacks like an asymptote,
 then ...
 It's an AFLAC commercial.
 

   copyright 2005 Bruce Kirkpatrick

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