Look at the following…
Exponential Functions
f ( x) = 4 x 2 − 3 x + 1
„
f ( x) = 4 x − 3
Polynomial
Exponential
Objective: To graph exponentials
equations and functions, and solve
applied problems involving exponential
functions and their graphs.
Real World Connection
„
Exponential functions are used to model
numerous real-world applications such
as population growth and decay,
compound interest, economics
(exponential growth and decay) and
more.
Exponential Function
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The function f(x) = bx, where x is a
real number, b > 0 and b ≠ 1, is
called the exponential function,
base b.
REVIEW
„
„
Remember: x0 = 1
__________ – slides a figure without
changing size or shape
Examples of
Exponential Functions
f ( x ) = 3x
⎛1⎞
f ( x) = ⎜ ⎟
⎝3⎠
x
f ( x) = (4.23) x
(The base needs to be positive in order to
avoid the complex numbers that would occur
by taking even roots of negative numbers.)
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Graphing Exponential Functions
Example
Use the function
„
Compute function values and list the
results in a table.
Plot the points and connect them with a
smooth curve. Be sure to plot enough
points to determine how steeply the curve
rises.
1.
f(x) = 13.49 (0.967) x – 1
to find the number of О-rings
О rings expected to fail at a
temperature of 60° F. Round to the nearest whole
number.
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2.
Graph the
exponential function
Graph the exponential function
y = f(x) = 3x.
x
⎛ 1⎞
y = f (x) = ⎜ ⎟
⎝ 3⎠
(0, 1)
0
1
(0, 1)
(1 3)
(1,
−1
1
3
( 1 3)
(−1,
x
y = f(x) = 3x
(x, y)
0
1
x
1
3
9
(2, 9)
−2
9
(−2, 9)
3
27
(3, 27)
−3
27
(−3, 27)
−1
1/3
(−1, 1/3)
1
1/3
(1, 1/3)
−2
1/9
(−2, 1/9)
2
1/9
(2, 1/9)
−3
1/27
(−3,1/27)
3
1/27
(3,1/27)
„
„
„
„
„
„
„
Domain = __________
Range = (0, ∞)
Passes through the point __________
If b>1, then graph goes up to the right and is
__________.
If __________, then graph goes down to the right
and is decreasing.
Graph is one-to-one and has an inverse.
Graph approaches but does not touch x-axis.
x
(x, y)
2
Characteristics of Exponential
Functions, f(x) = bx
⎛1⎞
y = f ( x) = ⎜ ⎟
⎝3⎠
Graphing an Exponential
Transformation
„
The graph of y=abx-h+ k is the
graph of y=abx translated h
units
it __________ and
d k units
it
vertically.
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Example -- Graph
Example:
y = 3x + 2.
The graph is a reflection of the graph of y = 3x across
the y-axis, followed by a reflection across the x-axis
and then a shift up of 4 units.
The graph is that of y = 3x shifted left 2 units.
x
y= 3 x+2
x
y
−3
3
1/3
−3
3
−23
23
−2
1
−2
−5
−1
3
−1
1
0
9
0
3
1
27
1
3.67
2
81
3
243
2
3.88
3
3.96
Observing Relationships
Connecting the Concepts
The number e
„
„
„
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The number e is an __________
number.
Value of e ≈ 2.71828
2 71828
Note: Base e exponential functions are
useful for graphing __________ growth
or decay.
Graphing calculator has a key for ex.
Graph y = 4 − 3−x
Practice with the Number e
„
Find each value of ex, to four decimal
places, using the ex key on a calculator.
a) e4
b) e−0.25
c) e2
Answers:
a) 54.5982
c) 7.3891
d) e−1
b) 0.7788
d) 0.3679
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Natural Exponential Function
f(x) = e
⎛ r⎞
A = P ⎜1 + ⎟
⎝ n⎠
x
Remember
e is a number
e lies between 2 and 3
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„
Compound Interest Formula
„
„
„
„
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Compound Interest Formula for
Continuous Compounding
A = Pe rt
„
„
„
„
nt
A = _________ in account after t years
P = principal amount of money _________
R = interest rate (__________ form)
N = number of __________ __________
interest is compounded
T = time in __________
Compound Interest Example
„
A sum of $10,000 is invested at an annual rate of 8%.
Find the balance in that account after 5 years subject to a)
quarterly compounding and b) continuous compounding.
A = amount in account after t years
P = principal amount of money invested
R = interest rate (decimal form)
T = time in years
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