Exponential Growth and Decay

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1 Exponential Growth and Decay

2 Identifying Exponential Growth vs Decay A. Exponential Equation: f(x) = Ca x 1. C: COEFFICIENT 2. a: BASE 3. X: EXPONENT B. Exponential Growth 1. When the base is greater than 1, a > 1 2. f(x) is INCREASING C. Exponential Decay 1. When the base is between zero and one, 0 < a < 1 2. f(x) is DECREASING 2

3 Example 1 Determine whether growth or decay. f x 1 5 x represents exponential Exponential Decay 3

4 Example 2 Determine whether growth or decay. f x x represents exponential Exponential Growth 4

5 Example 3 Determine whether growth or decay. f x x represents exponential Exponential Growth 5

6 Graphing Exponentials A. Make a table of values, usually from [ 3, 3] B. Plot the points from the table C. Connect the dots and draw from left to right, a smooth curve and label any asymptotes 6

7 Example 4 Graph f(x) = 2 x x f(x) (2) 3 (2) 2 (2) 1 (2) 0 (2) 1 (2) 2 (2) 3 1/8 1/4 1/

8 Example 4 Graph f(x) = 2 x x f(x) 3 (2) 3 1/8 2 (2) 2 1/4 1 (2) 1 1/2 0 (2) (2) (2) (2) 3 8 8

9 Example 4 Graph f(x) = 2 x (Calculator check) 9

10 Example 5 Graph f x x x f(x) 3 (1/3)(3) 2 1/81 2 (1/3)(3) 2 1/27 1 (1/3)(3) 1 1/9 0 (1/3)(3) 0 1/3 1 (1/3)(3) (1/3)(3) (1/3)(3)

Example 6 Graph f x 1 2 4 x x f(x) 3 (1/2)(4) 3 1/128 2 (1/2)(4) 2 1

11 Example 6 Graph f x x x f(x) 3 (1/2)(4) 3 1/128 2 (1/2)(4) 2 1 (1/2)(4) 1 1/32 1/ (1/2)(4) 0 (1/2)(4) 1 (1/2)(4) 2 (1/2)(4) 3 ½

12 Example 6 Graph f x 1 2 x x f(x) 3 (1/2) (1/2) (1/2) (1/2) (1/2) 1 1/2 2 (1/2) 2 1/4 3 (1/2) 3 1/8 12

Example 6 Graph f x x 1 2 x f(x) 3 (1/2) 3 8 2 (1/2) 2 4 1

13 Example 6 Graph f x x 1 2 x f(x) 3 (1/2) (1/2) (1/2) (1/2) (1/2) 1 1/2 2 (1/2) 2 1/4 3 (1/2) 3 1/8 13

14 Growth/Decay Factor Equation A. Equation: A = P (1 + r) t 1. P: Initial Principle 2. +: Growth Factor 3. -: Decay Factor 4. r: Interest Rate 5. t: Time it takes to accrue amount A P 1r t 14

15 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 P 1r A =? Do you know much you are going to make? P = $5,000 $5,000 is deposited r = 0.10 Interest Rate remember it needs to be in decimal form t = 3 Time it takes to accrue amount + Growth t 15

16 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 P 1r t A (0.10) 3 A $6,655 16

17 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 P 1r A =? Do you know much you are going to make? P = $5,000 $5,000 is spent r = 0.10 Interest Rate remember it needs to be in decimal form t = 3 Time it takes to accrue amount Decay Decreasing Value t 17

18 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 P 1r t A (0.10) 3 A $3,645 18

19 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? $4,

20 Example 10 A P 1r t You invest $5000 in an account that pays 6.25% interest per year. How much money will your investment be in 5 years? Exponential Functions 20

21 Example 10 A P 1r t 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 Exponential Functions 21

22 Example 10 A P 1r 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 A (0.0625) A A $ round to the nearest hundredths. t Exponential Functions 22

23 Example 11 A P 1r 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? t Exponential Functions 23

24 Example 11 A P 1r t 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 Exponential Functions 24

25 Example 11 A P 1r 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 Plug into the equation (0.0625) A Exponential Functions A $ round to the nearest hundredths. t 25

26 Example 12 A P 1r 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? t Exponential Functions 26

27 Example 12 A P 1r 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? A A P 1r t A $ t Exponential Functions 27

28 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) Exponential Functions 28

29 Compound Interest Equation r nt A P 1 n A = Total Amount Earned P = Principal r = Interest Rate n = Compounded Amount t = Time Exponential Functions 29

30 Example 13 A P1 r n nt $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 Exponential Functions 30

31 Example 13 A P1 r n nt $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 Exponential Functions 31

32 Example 13 A P1 r n 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. nt A Plug into the equation (4)(25) A $ round to the nearest hundredths Exponential Functions 32

33 Example 14 A P1 r n nt 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? Exponential Functions 33

34 Example 14 r A P1 n 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? nt P $ round to the nearest hundredths Exponential Functions 34

3.1 Exponential Functions and Their Graphs Date: Exponential Function

3.1 Exponential Functions and Their Graphs Date: Exponential Function Exponential Function: A function of the form f(x) = b x, where the b is a positive constant other than, and the exponent, x, is a variable.