8-2 properties of exponential functions
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82 Properties of Exponential Functions
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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.
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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
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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.
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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.
ACTIVITYComplete the 8.2 Exploration with a partner.
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Example #1: RealWorld ConnectionSuppose 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?
A = PertA = 1050 e(0.055 5)A = 1050 e(0.275)A = 1382.36A = $1382
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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
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Example #3: Evaluating ex
Graph y = ex. Evaluate e2 to four decimal places.
The value of e2 is about 7.3891.
The Number e
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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.
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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.
Comparing Graphs
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Example #5: Graphing y = abx for 0 < |a| < 1.
Graph each function and label the asymptote of each graph.
a. b.
x y x y
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Example #6: Graph each function.
a. y = 4(2)x b. y = 3x
x yx y
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Example #7: Translating y = abx.
Graph the stretch and then the translation .
x y x y
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Example #8: Graph the stretch y = 2(3)x and then each translation.
a. y = 2(3)x + 1
b. y = 2(3)x 4c. y = 2(3)x 3 1
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Example #3 RealWorld Connection
Example #9:
Number of 6 Hour Intervals 0 1 2 3 4 5 6 Number of Hours Elapsed 0 6 12 18 24 30 36 Technetium99m (mg)
100 50 25 12.5 6.25 3.13 1.56
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slide
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Hmwk: page 442(2 30 even, 40 47)