Algebra 2 Exponents as Variables
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Exponential Equations
Exponents as Variables

 

 We have graphed stuff like Y = X 2 ...

 

 

 
 How about Y = 2 X?
 
 Well, we know that when X = 0, Y = 1
 since anything to the zero power equals 1.
 
 We also know that when X = 1, Y = 2.
 Lets see what happens at even bigger X values ...
 
X Y
0 1
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024
 
 So it looks like this ...

 

 
 So for positive values of X anyway, 
 this sort of looks like the graph of Y = X 2
 I wonder which one goes up faster?
 
 Y = X2
X 0 1 2 3 4 5 6 7 8 9 10
Y 0 1 4 9 16 25 36 49 64 81 100
 
 Y = 2X
X 0 1 2 3 4 5 6 7 8 9 10
Y 1 2 4 8 16 32 64 128 256 512 1024
 
 Wow! no doubt about it.
 After a close start and some lead changes,
 Y = 2 X goes up much faster on the right side of graph.
 I wonder what happens on the left side of the graph of Y = 2 X?
 
 Remember that negative exponents don't mean negative numbers,
 they mean "put this term in the denominator"
 So:

 
 So as X values go off to the left,
 the denominator keeps getting bigger and Y keeps getting smaller.
 But Y always stays positive.
 That means that this equation (and most other exponential equations),
 has a horizontal asymptote at Y = 0.
 

 

 Don't forget that an asymptote is just a really fancy name
 for some value that a graph gets really close to.
 Here, our graph line gets closer and closer to Y = 0
 but never actually gets there.
 We KNOW that it never gets there because ...
 

 

 
 Way back when we learned a bunch of exponent rules.
 They still apply when the exponent is an X (or any other letter)
 and not a number.

 

 

   copyright 2005 Bruce Kirkpatrick

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