exponential functions and their graphs. 2 the exponential function f with base b is defined by f(x)...
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Exponential Functions and Their Graphs
2
The exponential function f with base b is defined by
f(x) = abx
where b > 0, b 1, and x is any real number.
** when b> 1; b is considered a growth factor.
For instance,
f(x) = 3x and g(x) = 0.5x
are exponential functions.
3
The value of f(x) = 3x when x = 2 is
f(2) = 32 =
The value of g(x) = 0.5x when x = 4 is
g(4) = 0.54 =
The value of f(x) = 3x when x = –2 is
9
1
9
f(–2) = 3–2 =
0.0625
4
The graph of f(x) = abx, b > 1
y
x(0, 1)
Domain: (–, )
Range: (0, )
Horizontal Asymptote y = 0
4
4
5
The graph of f(x) = abx, 0 < b < 1
y
x
(0, 1)
Domain: (–, )
Range: (0, )
Horizontal Asymptote y = 0
4
4
Since a< 1; a is decay factor.
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Example: Sketch the graph of f(x) = 2x.
x
y
2–2
2
4
x f(x)
-2 -1
0
1
2
2-2 = 1/4
2-1 = 1/2
20 = 1
21 = 2
22 = 4
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Example: Sketch the graph of g(x) = 2x – 1. State the domain and range.
x
yThe graph of this function is a vertical translation of the graph of f(x) = 2x
down one unit .
f(x) = 2x
y = –1 Domain: (–, )
Range: (–1, )
2
4
8
Example: Sketch the graph of g(x) = 2-x. State the domain and range.
x
yThe graph of this function is a reflection the graph of f(x) = 2x in the y-axis.
f(x) = 2x
Domain: (–, )
Range: (0, ) 2–2
4
9
Example: Sketch the graph of g(x) = 4x-3 + 3. State the domain and range.
x
yMake a table.
Domain: (–, )
Range: (3, ) or y > 3
2–2
4 x y
3 4
2 3.25
1 3.0625
4 7
5 19
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Write an exponential function y = abx for the graph that includes the given points.
(2, 2) and (3, 4)
y = abx Substitute in (2,2) for x and y.2 = ab2
Solve for a2 = ab2
Now substitute (3,4) in for x and y into y = abx
4 = ab3
b3
4 = a b3
Set them equal to each other:2 = 4 = a b3 b2
Now solve for b to get b=2
You must solve for a: 2/2^2 = (1/2) = aSubst you’re a = (1/2) & b = 2 into either one of your 2 equations: y = abx
Y = (1/2)(2)^x
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Write an exponential function y = abx for the graph that includes the given points.
(2, 4) and (3, 16)
y = abx
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The irrational number e, where
e 2.718281828…
is used in applications involving growth and decay.
Using techniques of calculus, it can be shown that
ne
n
n
as 1
1
The Natural Base e
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The graph of f(x) = ex
y
x2 –2
2
4
6
x f(x)
-2 0.14
-1 0.38
0 1
1 2.72
2 7.39
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Example: Sketch the graph of g(x) = ex-5 + 2. State the domain and range.
x
yMake a table.
Domain: (–, )
Range: (2, ) or y > 2
2–2
4 x y
5 3
6 4.72
7 9.39
4 2.36
3 2.14
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One to One Property
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
If the bases are the same, then set the exponents equal.
Example—3x = 32
x = ___
Example—3x+ 1 = 9
x = ___
Example—2x - 1 = 1/32
x = ___
Example—e3x + 2 = e3
x = ___
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Formulas for Compound Interest—
1.) compound per year -- A = P 1 + r nt
n
Interest Applications
Balance in account Principal ($ you invest)
r is the rate
n is the number times you compound your money per year
t is time.
2. Compounded continuously– A = Pert
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A total of $12000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded
a. quarterlyb. monthlyc. continuously