1. 6.4 Exponential Growth and decay

2. What We Will Learn Use and identify growth and decay functions
Interpret and rewrite functions
Story problems

3. Needed Vocab
Exponential growth: quantity increases by the same factor over equal intervals of time
Exp. Growth function: 𝑦=𝑎 1+𝑟 𝑡
Exponential decay: quantity decreases by the same factor over equal intervals of time
Exp decay function: 𝑦=𝑎 1−𝑟 𝑡

4. Ex. 1 Using Exp Growth Function
The inaugural attendance of an annual music festival is 150,000. The attendance y increases by 8% each year.
A. Write an exponential growth function
Use 𝑦=𝑎 1+𝑟 𝑡
Need a and r
a = initial amount
r = rate of growth or decay as a DECIMAL
t = time in years
𝑦=150, 𝑡
𝑦=150, 𝑡
B. How many people will attend in the 5th year?
Use function from last question
t = 4 because year one is always zero
𝑦=150,
𝑦=150,
𝑦=204,073

5. Your Practice
A website has 500,000 members in The numbers y of members increases by 15% each year. (a) Write an exponential function t years after (b) How many members will there be in 2016? Round to nearest ten thousand.
a. 𝑦=500, 𝑡
𝑦=500, 𝑡
b. 𝑦=500,
𝑦=500,
𝑦=1,160,000

6. Ex. 2 Identifying Growth and Decay
Look at y values
Multiplying is growth
Dividing (or multiplying by fraction) is decay
If neither, then answer is neither
Is it growth, decay, or neither?
Dividing by 3 or multiplying 1/3
Decay
Multiplying 2
Growth x y
270 1 90 2 30 3 10 X 1 2 3 Y 5 10 20 40

7. Exs. 3 and 4 Is It Growth or Decay
Look at (inside)
If inside > 1, then growth
If inside < 1, then decay
𝑦= 𝑡
Growth
𝑦= 𝑡−3
Decay

8. Ex. 5 Compound Interest
You deposit $100 in a savings account that earns 6% annual interest compounded monthly. Write a function that represents the balance after t years.
A = ?
P = 100
R = .06
N = 12
T = t
𝐴=𝑃 𝑡
𝐴= 𝑡
𝐴= 𝑡
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
A = amount you will have
P = amount you invest
R = interest rate as a decimal
N = number of times per year being compounded
T = time in years

9. Your Practice
You deposit $500 in a savings account that earns 9% annual interest compounded quarterly. Write a function that represents the balance y after t years.
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
A= a
P= 500
R= .09
N = 4
T = t
𝐴= 𝑡
𝐴= 𝑡
𝐴= 𝑡

10. Ex. 6 and 7 Story Problems Going to use one of 3 equations
The value of a car is $21,500. It loses 12% of its value every year. Write a function that represents the value of the car in t years. Find the approximate value of the car after 6 years.
Loses: so decay
𝑦=𝑎 1−𝑟 𝑡
𝑦= −.12 𝑡
𝑦= 𝑡 𝑦= 𝑦=
𝑦=$
Going to use one of 3 equations
1. 𝑦=𝑎 1+𝑟 𝑡
Growth
2. 𝑦=𝑎 1−𝑟 𝑡
Decay
3. 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
Compound interest