Algebra 2 Vertical Asymptotes
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You Can't Go There
Vertical Asymptotes

 

 Look at this equation:

 

 
 This looks kind of simple,
 but look what happens when X = 1 ...

 

 
 We have a big problem. 
 We can't deal with a zero in the denominator.
 That means that X can't equal 1.
 
 Let's make a T table to see what happens to Y
 when X gets close to 1.
 
X Y
0 -1
.5 -2
.75 -4
.9 -10
.95 -20
.99 -100
1.01 100
1.05 20
1.1 10
1.25 4
1.5 2
2 1
 
 Now we can draw the graph.
 Since X can't equal 1, draw a dotted line where X = 1 ...
 

 

 
 Now we put the X, Y points from the T table into the graph.
 Look at what happens when the graph gets close to X = 1 ...

 

 
 When X gets close to X = 1 from either side
 the graph line shoots off the drawing.
 
 When X is smaller than 1, 
 the values of Y get to be really big negative numbers
 the closer we get to X = 1.
 
 When X is bigger than 1, 
 the values of Y get to be really big positive numbers
 the closer we get to X = 1.
 
 Math people really get excited about strange stuff like this.
 They usually give strange stuff strange names.
 That's what they did here.
 They call lines like the one at X = 1, an asymptote.
 Since the line goes up and down, they call it a vertical asymptote.
 A vertical asymptote only happens 
 at an X value that makes the denominator equal zero,
 that we can't get rid of by simplifying the fraction.
 
 Examples:
 

 

 
 Can we have more than one vertical asymptote
 in the same equation???
 
 SURE!
 
 We can have lots!
 
 Here's an equation with two:
 

 

 
 This equation will have a vertical asymptote where X = -1,
 and another one where X = 1.
 

 

 
 Let's build a T table and see what this puppy looks like ...
 
X Y
-3 .125
-2 .333
-1.5 .8
-1.1 4.76
-1.01 49.8
-0.99 -50.2
-0.9 -5.02
0 -1
0.9 -5.02
0.99 -50.2
1.01 49.8
1.1 4.76
1.5 .8
2 .333
3 .125
 
 Now draw the graph ...

 

 

   copyright 2005 Bruce Kirkpatrick

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