4.3 Exponential Functions 2011
April 22, 2011
4.3 Exponential Functions
Objectives:
Solve exponential equations.
Graph exponential functions.
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4.3 Exponential Functions 2011
April 22, 2011
Review of Exponent Rules
Let s, t, a, and b all be real numbers
and a > 0 and b > 0.
as at = as+t a2 a4 = a6
32 33 = 35 = 243 (as)t = ast
(a2)4 = a8
(32)3 = 36 = 729
(ab)s = as bs
(3a)5 = 35a5 = 243a5
a­s = 1/as 3­5 = 1/35 = 1/243 (1/3)­5 = 1/3­5 = 35 1s = 1
15 = 1
125418965 = 1
a0 = 1 9,254,135,254,1550 = 1
520 = 1
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4.3 Exponential Functions 2011
April 22, 2011
Solve Exponential Equation
x + 1
3 = 81
x + 1
4
3 = 3
x + 1 = 4
x = 3
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4.3 Exponential Functions 2011
Solve for x.
April 22, 2011
­x
x .
e = (e )
2
2
1
e3
e = e 2x . e ­3
­x2
2
2x ­ 3
­x
e = e ­x2 = 2x ­ 3
2
0 = x + 2x ­ 3
0 = (x + 3)(x ­ 1)
x = ­3
x = 1
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4.3 Exponential Functions 2011
April 22, 2011
Properties of the Exponential Function
f(x) = ax a > 1 1) Domain and Range
D: x = all real #s
R: y > 0
2) Intercepts
no x intercepts y intercept is 1 (0, 1)
3) Horizontal Asymptote
HA: y = 0
(as x ⇒-∞)
4) Categorizing the Function
Increasing Function
One­to­one
5) Points on the Graph
(0, 1) (1, a) (­1, 1/a)
(1, a)
(0, 1)
(­1, 1/a)
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4.3 Exponential Functions 2011
April 22, 2011
Properties of the Exponential Function
f(x) = ax 0 < a < 1
1) Domain and Range
D: x = all real #s
R: y > 0
2) Intercepts
no x intercepts y intercept is 1 (0, 1)
3) Horizontal Asymptote
(as x ⇒∞)
HA: y = 0
4) Categorizing the Function
Decreasing Function
One­to­one
5) Points on the Graph
(0, 1) (1, a) (­1, 1/a)
(­1, 1/a)
(0, 1)
(1, a)
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4.3 Exponential Functions 2011
April 22, 2011
Lets try together!
f(x) = 3x
The base is 3. So a = 3.
So we know that the points (1, 3), (0,1), (­1,1/3).
(1,3)
(­1,1/3)
(0,1)
(1,3)
(­1,1/3)
(0,1)
Now we can draw the line.
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4.3 Exponential Functions 2011
1
f(x) = ( /4)
April 22, 2011
Lets try together!
x
The base is 1/4. So a = 1/4.
So we know that the points (1, 1/4), (0,1), (­1,4).
(­1,4)
(0,1)
(1,1/4)
(0,1)
Now we can draw the line.
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4.3 Exponential Functions 2011
April 22, 2011
Lets try together!
­x
f(x) = 2 ­ 3
The base is 2. So a = 2.
So we know that the points (1, 2), (0, 1), (­1, 1/2).
(1, 2)
(­1, 1/2)
(0, 1)
(0, 1)
Now we can draw the line.
Now we can graph the transformations
on the next page.
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4.3 Exponential Functions 2011
April 22, 2011
Lets try together!
­x
f(x) = 2 ­ 3
f(x) = 2­x ­ 3 flips the graph
about the y­axis.
So we know that the points (­1, 2), (0, 1), (1, 1/2).
(­1, 2)
(0, 1)
(1, 1/2)
f(x) = 2­x ­ 3 shifts the graph down 3 units.
(­1, ­1)
(0, ­2)
(1, ­21/2)
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4.3 Exponential Functions 2011
April 22, 2011
Homework: page 282
(6 - 10, 29 - 36, 53 - 66, 72, 73, 75, 78, 79, 98)
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