Math 110
Section 4.3
An exponential function is a function f, such that
where a is an actual number
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f ( x)  a , a  0, a  1
f ( x)  10 x
f ( x)  2 x
Examples:
1
f ( x)   
 2
Observations:
Domain =
Range =
x
x
 , 
0, 
 3
f ( x)   
 4
x
(you can choose any value you want for x)
(since a must be positive, the output must be positive, too)
If you graph the function, the graph will go through
1

  1, 
a

f ( x)  a x

f (1)  a 1
0,1
f ( x)  a x

f (0)  a 0

f ( 0)  1
1, a
f ( x)  a x

f (1)  a1

f (1)  a

f ( 1) 
1
a
For a > 1
The graph is nearly horizontal as you go left (negative)
Since you're making a smaller and
smaller fraction
But you never actually hit the y axis
It's increasingly steep as you go rightward
Since you're making a larger and
larger number
For 0 < a < 1
The graph is nearly horizontal as you go right
Since you're making a smaller and
smaller fraction
But you never actually hit the y axis
It's increasingly steep as you go leftward
Since you're making a larger and
larger number
Notice that
1
f ( x)   
 2
Math 110
x
looks just like
f ( x)  2 x , except reflected across the y axis
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Math 110
Section 4.3
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<Draw a couple>
ICE: Have them draw a couple
Math 110
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Math 110
Section 4.3
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Transformations
Negatives can be in two places: in the exponent, and in front of the base:
2x
x
in front of the base  2
in the exponent:
Effects:
x
In the exponent: 2
This actually reflects it across the y axis
In front of the base  2
Reflects is across the x axis, since we’re taking the normal output, and making it
negative
x
Do demo:
 2 x  3  1
Properties Useful for Solving Exponential / Logarithmic Equations
1. If
x y
x
y
a

a
If
then
ax  ay
then
x y
Example:
5 x  125  5 x  5 3 x = 3
5 x 1  125  5 x 1  5 3 x -1 = 3  x = 4
etc
Math 110
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