8­2 Properties of Exponential Functions
March 24, 2009
8­2 Properties of Exponential Functions
Objectives:
• Determine the future value of an investment if the interest is compounded continuously.
• Use e as a base.
• Identify the role of the constants in y = abcx.
Mar 9­9:36 AM
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8­2 Properties of Exponential Functions
March 24, 2009
Write an equation for each translation.
1. y = |x|
1 unit up, 2 units left
2. y = ­|x|
2 units down
3. y = x2
2 units down, 1 unit right
4. y = ­x2
3 units up, 1 unit left
Mar 9­9:38 AM
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8­2 Properties of Exponential Functions
March 24, 2009
Write each equation in simplest form.
Assume that all variables are positive.
8. Use the formula for simple interest I = Prt. Find the interest for a principal of $550 at a rate of 3% for 2 years.
Mar 9­10:17 AM
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8­2 Properties of Exponential Functions
March 24, 2009
Yesterday, you studied simple and compound interest. The more frequently interest is compounded, the more quickly the amount in an account increases. The formula for continuously compounded interest uses the number e.
ACTIVITY
Complete the 8.2 Exploration with a partner.
Mar 9­9:56 AM
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8­2 Properties of Exponential Functions
March 24, 2009
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8­2 Properties of Exponential Functions
March 24, 2009
Example #1: Real­World Connection
Suppose you invest $1050 at an annual interest rate of 5.5% compounded continuously. How much money, to the nearest dollar, will you have in the account after five years?
rt
A = Pe
A = 1050 e(0.055 5)
A = 1050 e(0.275)
A = 1382.36
A = $1382
Mar 9­2:12 PM
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8­2 Properties of Exponential Functions
March 24, 2009
Example #2:
Suppose you invest $1300 at an annual interest rate of 4.5% compounded continuously. How much money, to the nearest cent, will you have in the account after three years?
A = Pert
Mar 9­9:57 AM
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8­2 Properties of Exponential Functions
March 24, 2009
The Number e
Example #3: Evaluating ex
Graph y = ex. Evaluate e2 to four decimal places.
The value of e2 is about 7.3891.
Mar 9­9:53 AM
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8­2 Properties of Exponential Functions
March 24, 2009
Example #4: Use the graph of y = ex to evaluate each expression to four decimal places.
You can also use the e button on your calculator to evaluate each expression.
a. e4
b. e­3
c. Mar 9­9:56 AM
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8­2 Properties of Exponential Functions
March 24, 2009
Comparing Graphs
The function f(x) = bx is the parent of a family of exponential functions for each value of b. The factor a in y = abx stretches, shrinks, and/or reflects the parent.
Mar 9­9:42 AM
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8­2 Properties of Exponential Functions
March 24, 2009
Example #5: Graphing y = abx for 0 < |a| < 1.
Graph each function and label the asymptote of each graph.
a.
x
y
b.
x
y
Mar 9­9:43 AM
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8­2 Properties of Exponential Functions
March 24, 2009
Example #6: Graph each function.
a. y = ­4(2)x
x
y
b. y = ­3x
x
y
Mar 9­9:50 AM
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8­2 Properties of Exponential Functions
March 24, 2009
Example #7: Translating y = abx.
Graph the stretch x
y
and then the translation
x
.
y
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8­2 Properties of Exponential Functions
March 24, 2009
Example #8: Graph the stretch y = 2(3)x and then each translation.
a. y = 2(3)x + 1
b. y = 2(3)x ­ 4
c. y = 2(3)x ­ 3 ­ 1
Mar 9­9:51 AM
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8­2 Properties of Exponential Functions
March 24, 2009
Example #9: Number of 6 Hour Intervals 0 1 2 3 4 5 6 Example #3 Real­World Connection
Number of Hours 0 6 12 18 24 30 36 Elapsed Technetium­99m (mg) 100 50 25 12.5 6.25 3.13 1.56 Mar 9­9:52 AM
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8­2 Properties of Exponential Functions
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slide
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8­2 Properties of Exponential Functions
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8­2 Properties of Exponential Functions
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Hmwk: page 442
(2 ­ 30 even, 40 ­ 47)
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