6.4 Exponential Growth and decay

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

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

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=150,000 1+.08 𝑡 𝑦=150,000 1.08 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=150,000 1.08 4 𝑦=150,000 1.36 𝑦=204,073

Your Practice A website has 500,000 members in 2010. The numbers y of members increases by 15% each year. (a) Write an exponential function t years after 2010. (b) How many members will there be in 2016? Round to nearest ten thousand. a. 𝑦=500,000 1+.15 𝑡 𝑦=500,000 1.15 𝑡 b. 𝑦=500,000 1.15 6 𝑦=500,000 2.313 𝑦=1,160,000

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

Exs. 3 and 4 Is It Growth or Decay Look at (inside) If inside > 1, then growth If inside < 1, then decay 𝑦=5 1.07 𝑡 Growth 𝑦=6 .86 𝑡−3 Decay

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 1+ .06 12 12𝑡 𝐴=100 1+.005 12𝑡 𝐴=100 1.005 12𝑡 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

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=500 1+ .09 4 4𝑡 𝐴=500 1+.0225 4𝑡 𝐴=500 1.0225 4𝑡

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=21500 1−.12 𝑡 𝑦=21500 .88 𝑡 𝑦=21500 .88 6 𝑦=21500 .4644 𝑦=$9984.69 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