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Chapter 7 – Exponential and logarithmic functions
7.1 – Exploring Exponential Models
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7.1 – Exploring Exponential Models
#Goals: To model exponential growth and decay
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7.1 – Exploring Exponential Models
Definition A function with the general form y = abx Facts/Characteristics a ≠ 0 b > 0 b ≠ 1 Examples Non-Examples Exponential Function
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7.1 – Exploring Exponential Models
Example 1 What is the graph of y = 3.1x
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7.1 – Exploring Exponential Models
Got it 1? What is the graph of y = 2(3)x
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7.1 – Exploring Exponential Models
Definition As the value of x increases, the value of y increases Facts/Characteristics Examples Non-Examples Exponential Growth
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7.1 – Exploring Exponential Models
Definition As the value of x increases, the value of y decreases, approaching 0 Facts/Characteristics Examples Non-Examples Exponential Decay
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7.1 – Exploring Exponential Models
Definition A line that a graph approaches as x or y increases in absolute value Facts/Characteristics Examples Non-Examples Asymptote
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7.1 – Exploring Exponential Models
Exponential Function For the function y = abx a > 0 b > 1 Exponential Growth 0 < b < 1 Exponential Decay Domain: all reals y-intercept: (0, a) Range: {y|y > 0} Asymptote: y = 0
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7.1 – Exploring Exponential Models
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7.1 – Exploring Exponential Models
Example 2 Identify y = 0.7x as an example of exponential growth or decay. What is the y-intercept?
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7.1 – Exploring Exponential Models
. Got it: What is the graph of y = 4( )x
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7.1 – Exploring Exponential Models
Definition The value of b when b > 1 Facts/Characteristics Examples Non-Examples Growth Factor
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7.1 – Exploring Exponential Models
Definition The value of b when 0 < b < 1 Facts/Characteristics Examples Non-Examples Decay Factor
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7.1 – Exploring Exponential Models
Growth Factor Increases by constant percentage each time period percent increase r rate of increase growth rate b = 1 + r Decay Factor Decreases by constant percentage each time period percentage decrease rate of decay
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7.1 – Exploring Exponential Models
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7.1 – Exploring Exponential Models
Example 3 You buy a savings bond for $25 that pays a yearly interest rate of 4.2%. What will the savings bond be worth after fifteen years?
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7.1 – Exploring Exponential Models
Got it 3? Suppose you deposit $2000 in a savings account that pays interest at an annual rate of 4%. If no money is added or withdrawn from the account, how much will be in the account after 18 years?
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7.1 – Exploring Exponential Models
Example 4 You open a savings account that pays 4.5% annual interest. If your initial investment is $300 and you make no additional deposits or withdrawals, how many years will it take for the account to grow to at least $500?
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7.1 – Exploring Exponential Models
Got it 4? Suppose you deposit $2000 in a savings account that pays interest at an annual rate of 4%. If no money is added or withdrawn from the account, how many years will it take for the account to contain $3000?
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7.1 – Exploring Exponential Models
Exponential functions are often discrete (individual points) There is never exactly $3000 in the account To model a discrete situation using an exponential function in the form y = abx, you need to find the growth or decay factor b If you know y-values for 2 consecutive x values, you can find the rate of change (r) and then find b and b = 1 + r
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7.1 – Exploring Exponential Models
Example 5 The initial value of a car is $30,000. After one year, the value of the car is $20,000. Estimate the value of the car after five years.
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7.1 – Exploring Exponential Models
Got it 5? A population of 1,860,000 decreases 1.5% each year for 12 years. Write an exponential function to model the situation. Find the amount after 12 years.
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7.1 – Exploring Exponential Models
HOMEWORK Page 439 #10 – 28 (even)
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