Exponential Growth and Decay

1
INTRODUCTION

• THE FOLLOWING PRESENTATION REQUIRES LITTLE OR NO
  FAMILIARITY WITH POWERPOINT.
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2
The Exponential Function
3
EXPONENTSA quick review
FACTOR
BASE
EXPONENT
Each 2 is called a
Finally, evaluate the expression
The 5 is called the
The 2 is called the
22222
25
32
Click here for practice problems on exponents.
Factored form
Exponential form
4
The Exponential Function
2x
f(x)
3
Example ?
Note the following
1. The variable now appears as an exponent.
2. The base must be a positive constant other
than 1.
3. A multiplier often appears in the formula.
5
The Graph of the Exponential Function
f(x) 2x
f(x) 2x
Example
f(x)
?
-3
-3
1/8
We will construct a table of values
-2
1/4
-1
1/2
1

?
?
1
0
2
1
?
4
2
f(x) 2x
?
?
3
8
?
? Place the base 2 in the denominator of a
fraction, and change the exponent to a positive
3
?
x
?
16
4
6
The Graph of the Exponential Function
Click here for practice problems on graphing the
exponential function.
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Properties of the Exponential Function
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1. The domain is all real numbers ? (-? lt x lt
?).
all possible values of x
Note that there is no restriction on the values
the exponent can assume
In fact, the values we chose for x included
negative and positive numbers, as well as 0
f(x) 2x
Also note how the graph stretches across the
entire x-axis
All real numbers
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Properties of the Exponential Function
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2. The range is all positive real numbers (f(x)
gt 0).
all possible values of y (or f(x))
Therefore, the entire graph of the function lies
above the x-axis (where f(x) gt 0)
All of the values produced by the function
formula are positive
All positive real numbers
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Properties of the Exponential Function
3. The x-axis is a horizontal asymptote.
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an imaginary line which the graph approaches,
but never reaches
Therefore, the graph has no x-intercept.
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a point where a graph intersects an axis
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Properties of the Exponential Function
4. The y-intercept is 1.
(0, 1)

11
Properties of the Exponential Function
5. The function is one-to-one.

• Like any other function, the exponential function
  passes the Vertical Line Test.

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f(x) 2x

• The function also passes the Horizontal Line
  Test.
• Such functions are called one-to-one.

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• Since the exponential function is 1-to-1, it has
  an inverse. This inverse function is the
  Logarithmic Function y log2 x.

f -1(x) log2 x




a new function, notated f -1(x), derived from
the original function by exchanging x and y its
graph is symmetrical to f(x) about the line y x.
No horizontal line intersects the graph more than
once
No vertical line intersects the graph more than
once





y x (the axis of symmetry)
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Properties of the Exponential Function
y
x
Reorienting the graphboard makes the symmetry of
the function and its inverse even more evident
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The End
The End
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