1. Exponential Growth and Decay

2. Identifying Exponential Growth vs Decay
Exponential Equation: f(x) = Cax
C: COEFFICIENT
a: BASE
X: EXPONENT
Exponential Growth
When the base is greater than 1, a > 1
f(x) is INCREASING
Exponential Decay
When the base is between zero and one, 0 < a < 1
f(x) is DECREASING
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7.1 & Graphing Exponential Transformations

3. 7.1 & 7.2 - Graphing Exponential Transformations
Example 1
Determine whether represents exponential growth or decay.
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7.1 & Graphing Exponential Transformations

4. 7.1 & 7.2 - Graphing Exponential Transformations
Example 2
Determine whether represents exponential growth or decay.
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7.1 & Graphing Exponential Transformations

5. 7.1 & 7.2 - Graphing Exponential Transformations
Example 3
Determine whether represents exponential growth or decay.
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7.1 & Graphing Exponential Transformations

6. Graphing Exponentials
Make a table of values, usually from [–3, 3]
Plot the points from the table
Connect the dots and draw from left to right, a smooth curve and label any asymptotes
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7.1 & Graphing Exponential Transformations

7. 7.1 & 7.2 - Graphing Exponential Transformations
Example 4
Graph f(x) = 2x x
f(x) –3
(2)–3
1/8 –2
(2)–2
1/4 –1
(2)–1
1/2
(2)0 1
(2)1 2
(2)2 4 3
(2)3 8 x
f(x) –3
(2)–3 –2
(2)–2 –1
(2)–1
(2)0 1
(2)1 2
(2)2 3
(2)3 x
f(x) –3 –2 –1 1 2 3
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7.1 & Graphing Exponential Transformations

8. 7.1 & 7.2 - Graphing Exponential Transformations
Example 4
Graph f(x) = 2x x
f(x) –3
(2)–3
1/8 –2
(2)–2
1/4 –1
(2)–1
1/2
(2)0 1
(2)1 2
(2)2 4 3
(2)3 8
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7.1 & Graphing Exponential Transformations

9. 7.1 & 7.2 - Graphing Exponential Transformations
Example 4
Graph f(x) = 2x (Calculator check)
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7.1 & Graphing Exponential Transformations

10. 7.1 & 7.2 - Graphing Exponential Transformations
Example 5
Graph x
f(x) –3 –2 –1 1 2 3 –3
(1/3)(3)–3
1/81 –2
(1/3)(3)–2
1/27 –1
(1/3)(3)–1
1/9
(1/3)(3) 0
1/3 1
(1/3)(3) 1 2
(1/3)(3) 2 3
(1/3)(3) 3 9 –3
(1/3)(3)–2 –2 –1
(1/3)(3)–1
(1/3)(3) 0 1
(1/3)(3) 1 2
(1/3)(3) 2 3
(1/3)(3) 3
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7.1 & Graphing Exponential Transformations

11. 7.1 & 7.2 - Graphing Exponential Transformations
Example 6
Graph x
f(x) –3 –2 –1 1 2 3 –3
(1/2)(4)–3
1/128 –2
(1/2)(4)–2
1/32 –1
(1/2)(4)–1
1/8
(1/2)(4) 0 1
(1/2)(4) 1 2
(1/2)(4) 2 8 3
(1/2)(4) 3 32
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7.1 & Graphing Exponential Transformations

12. 7.1 & 7.2 - Graphing Exponential Transformations
Example 6
Graph x
f(x) –3
(1/2)–3 8 –2
(1/2)–2 4 –1
(1/2)–1 2
(1/2)0 1
(1/2)1
1/2
(1/2)2
1/4 3
(1/2)3
1/8 x
f(x) –3
(1/2)–3 –2
(1/2)–2 –1
(1/2)–1
(1/2)0 1
(1/2)1 2
(1/2)2 3
(1/2)3 x
f(x) –3 –2 –1 1 2 3
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7.1 & Graphing Exponential Transformations

13. 7.1 & 7.2 - Graphing Exponential Transformations
Example 6
Graph x
f(x) –3
(1/2)–3 8 –2
(1/2)–2 4 –1
(1/2)–1 2
(1/2)0 1
(1/2)1
1/2
(1/2)2
1/4 3
(1/2)3
1/8
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7.1 & Graphing Exponential Transformations

14. Growth/Decay Factor Equation
Equation: A = P (1 + r)t
P: Initial Principle
+: Growth Factor
-: Decay Factor
r: Interest Rate
t: Time it takes to accrue amount
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7.1 & Graphing Exponential Transformations

15. 7.1 & 7.2 - Graphing Exponential Transformations
Example 7
You invest $5,000 in an account that pays 10% interest per year. How much money will your investment be in 3 years?
A = ? Do you know much you are going to make?
P = $5,000 $5,000 is deposited
r = Interest Rate – remember it needs to be in decimal form
t = 3 Time it takes to accrue amount
+ Growth
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7.1 & Graphing Exponential Transformations

16. 7.1 & 7.2 - Graphing Exponential Transformations
Example 7
You invest $5,000 in an account that pays 10% interest per year. How much money will your investment be in 3 years?
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7.1 & Graphing Exponential Transformations

17. 7.1 & 7.2 - Graphing Exponential Transformations
Example 8
You buy a car that cost $5,000 and depreciates 10% per year. What is the value of the car after 3 years?
A = ? Do you know much you are going to make?
P = $5,000 $5,000 is spent
r = Interest Rate – remember it needs to be in decimal form
t = 3 Time it takes to accrue amount
Decay Decreasing Value
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7.1 & Graphing Exponential Transformations

18. 7.1 & 7.2 - Graphing Exponential Transformations
Example 8
You buy a car that cost $5,000 and depreciates 10% per year. What is the value of the car after 3 years?
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7.1 & Graphing Exponential Transformations

19. 7.1 & 7.2 - Graphing Exponential Transformations
Example 9
You buy a car that cost $10,000 and depreciates 10% per year. How much money have you lost, compared to the original amount, in 5 years?
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7.1 & Graphing Exponential Transformations

20. 7.1 - Exponential Functions
Example 10
You invest $5000 in an account that pays 6.25% interest per year. How much money will your investment be in 5 years?
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7.1 - Exponential Functions

21. 7.1 - Exponential Functions
Example 10
You invest $5000 in an account that pays 6.25% interest per year. How much money will your investment be in 5 years?
A = ? Do you know much you are going to make?
P = $5,000 $5,000 is deposited
r = 6.25% Interest Rate – remember it needs to be in decimal form
t = 5 Time it takes to accrue amount
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7.1 - Exponential Functions

22. 7.1 - Exponential Functions
Example 10
You invest $5000 in an account that pays 6.25% interest per year. How much money will your investment be in 5 years?
Plug into the equation
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7.1 - Exponential Functions

23. 7.1 - Exponential Functions
Example 11
You buy a car that cost $5,000 and depreciates 6.25% per year. How much money will you end up paying in 5 years?
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7.1 - Exponential Functions

24. 7.1 - Exponential Functions
Example 11
You buy a car that cost $5,000 and depreciates 6.25% per year. How much money will you end up paying in 5 years?
A = ? Do you know much you are going to pay?
P = $5,000 $5,000 is borrowed
r = 6.25% Interest Rate – remember it needs to be in decimal form
t = 5 Time it takes to accrue amount
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7.1 - Exponential Functions

25. 7.1 - Exponential Functions
Example 11
You buy a car that cost $5,000 and depreciates 6.25% per year. How much money will you end up paying in 5 years?
Plug into the equation
7.1 - Exponential Functions
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26. 7.1 - Exponential Functions
Example 12
The value of a $3000 computer decreases about 30% each year. Write a function for the computer’s value in 4 years. Does the function represent growth or decay?
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7.1 - Exponential Functions

27. 7.1 - Exponential Functions
Example 12
The value of a $3000 computer decreases about 30% each year. Write a function for the computer’s value in 4 years. Does the function represent growth or decay?
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7.1 - Exponential Functions

28. 7.1 - Exponential Functions
Time Values
Annually is a one-time payment per year
Semiannually/Biannually is a payment every six months (2 times a year)
Quarterly is a payment every three months (4 times a year)
Monthly is a payment every month (12 times a year)
Daily is a payment every day (365 times a year)
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7.1 - Exponential Functions

29. Compound Interest Equation
A = Total Amount Earned
P = Principal
r = Interest Rate
n = Compounded Amount
t = Time
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7.1 - Exponential Functions

30. 7.1 - Exponential Functions
Example 13
$5,000 is deposited in an account that pays 6% annual interest compounded quarterly. Find the balance after 25 years if the interest is compounded quarterly.
A = ? Do we know how much it is when the balance after 25 years?
P = $5,000 $5,000 is deposited
r = 6% Interest Rate – remember it needs to be in decimal form
n = 4 Compounded quarterly
t = 25 Time it takes to accrue amount
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7.1 - Exponential Functions

31. 7.1 - Exponential Functions
Example 13
$5,000 is deposited in an account that pays 6% annual interest compounded quarterly. Find the balance after 25 years if the interest is compounded quarterly.
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7.1 - Exponential Functions

32. 7.1 - Exponential Functions
Example 13
Assume $5,000 is deposited in an account that pays 6% annual interest compounded quarterly. Find the balance after 25 years if the interest is compounded quarterly.
Plug into the equation
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7.1 - Exponential Functions

33. 7.1 - Exponential Functions
Example 14
How much must you deposit in an account that pays 6.5% interest, compounded quarterly, to have a balance of $5,000 in 15 years?
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7.1 - Exponential Functions

34. 7.1 - Exponential Functions
Example 14
How much must you deposit in an account that pays 6.5% interest, compounded quarterly, to have a balance of $5,000 in 15 years?
10/28/2019 1:24 AM
7.1 - Exponential Functions