Algebra 2 Removable Zeros
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Removable Zeros

 

 In general, when the number that we put in for X

 makes the denominator of a fraction be zero,
 we get a vertical (up and down) asymptote.
 
 But not always ...
 
 Say we have the equation:

 

 
 When X = 1 the denominator is zero.
 
 BUT WAIT!!!
 
 Before we do anything, lets try to simplify this.
 We can factor the top part, X 2 - 1 factors to (X + 1) times (X - 1).
 That means:

 

 
 and this we can simplify ...

 

 
 We can graph Y = X + 1 ...

 

 
 This is the graph of a straight line.
 But in the original equation, 
 the denominator was zero when X = 1!
 
 So what do we do? What does the graph of 
 

 

 
 look like?
 
 Does it have an asymptote where X = 1 or what?
 
 Here's the deal.
 
 If we have X values where the denominator equals zero,
 but we can factor away those places.
 The graph DOES NOT have an asymptote at these places.
 Graph the simplified equation BUT at the point where the denominator is zero,
 the graph has a hole.
 It sounds harder than it really is, watch ...
 
 We have the equation:

 

 We factor it to:

 

 That simplifies to:

 

 Graph Y = X + 1

 

 Put a hole in the graph at the place where the denominator was zero.

 

 
 Example:

 

 This factors to:

 

 This simplifies to:

 

 We can graph:

 

 
 In the original equation,
 the denominator equals zero when X = 0.
 That means the graph has a hole where X = 0.
 

 

 

   copyright 2005 Bruce Kirkpatrick

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