Section 7.6 Day 1 Exponential Growth & Decay Algebra 1

Compounded Interest Equation 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 Time r: interest rate n: number of times interest is compounded each year Initial Amount

Terminology Annually : Once a year Semi-annually: twice a year Quarterly: 4 times a year Monthly: 12 times a year Weekly: 52 times a year

Example 1 Maria’s parents invested $14,000 at 6% per year compounded monthly. How much money will there be in the account after 10 years? P: 14,000 R: 0.06 T: 10 N: 12 About 25, 471.55

Example 2 You invest $700 into an account with 3.5% interest for 20 years. Make a chart to calculate the following. Then, write a sentence on the results. Compounded annually Compounded semi-annually Compounded quarterly Compounded monthly Compounded weekly

Example 2 Compounded annually Compounded semi-annually 𝐴=700 1+ 0.035 1 20 1 =1392.85 Compounded semi-annually 𝐴=700 1+ 0.035 2 20 2 =1401.12 Compounded quarterly 𝐴=700 1+ 0.035 4 20 4 =1405.34 Compounded monthly 𝐴=700 1+ 0.035 12 20 12 =1408.19 Compounded weekly 𝐴=700 1+ 0.035 52 20 52 =1409.30

Exponential Growth General Equation Exponential Growth Equation: 𝑦=𝑎 1+𝑟 𝑡 Time Rate of Change (in decimal) Initial Amount

Exponential Decay General Equation Exponential Decay Equation: 𝑦=𝑎 1−𝑟 𝑡 Time Rate of Change (in decimal) Initial Amount

Exponential Growth: Example 1 PG 432 EX 1 The prize for a radio station contest begins with a $100 gift card. Once a day, a name is announced. The person has 15 minutes to call or the prize increase by 2.5% for the next day. A) What is the initial amount? B) What is the rate of change in a decimal? C) What is the time? D) What is the equation representing this situation? E) How much will the gift card be worth if no one wins after 10 days?

Exponential Growth: Example 1 Answers A) What is the initial amount? $100 B) What is the rate of change in a decimal? 0.025 C) What is the time? t D) What is the equation representing this situation? 𝑦=100 1.025 𝑡 E) How much will the gift card be worth if no one wins after 10 days? 𝑦=100 1.025 10 =$128.01

Exponential Decay: Example 2 PG 433 EX 3 A fully inflated child’s raft for a pool is losing 6.6% of its air every day. The raft originally contained 4500 cubic inches of air. A) What is the initial amount? B) What is the rate of change in a decimal? C) What is the time? D) What is the equation representing this situation? E) Estimate the amount of air in the raft after 7 days

Exponential Decay: Example 2 Answers A) What is the initial amount? 4500 cubic inches B) What is the rate of change in a decimal? 0.066 C) What is the time? t D) What is the equation representing this situation? 𝑦=4500 1−0.066 𝑡 E) Estimate the amount of air in the raft after 7 days 𝑦=4500 1−0.066 7 =2790.23 𝑐𝑢𝑏𝑖𝑐 𝑖𝑛𝑐ℎ𝑒𝑠

Example 3 In 2000, 2200 students attended Polaris High School. The enrollment has been declining 2% annually. How many students will be enrolled in 2015?

Example 3- Answer In 2000, 2200 students attended Polaris High School. The enrollment has been declining 2% annually. How many students will be enrolled in 2015? 𝑦=2200 1−0.02 15 =1624 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠

Sage & Scribe Find one partner Decide who is going to be person 1 and who is going to be person 2 Indicate that on your piece of paper and write your name down This will be collected for classwork points!

Partner 1: Partner 1: You are the Sage. You need to verbally explain how to solve the problem step by step. You CANNOT write anything Partner 2: You are the scribe. You CANNOT say anything and can ONLY write down EXACTLY what the Sage tells you to write down.

Example 4 Ms. Acosta received a job as a teacher with a starting salary of $34,000. According to her contract, she will receive a 1.5% increase in her salary every year. How much will Ms. Acosta earn in 7 years?

Example 4 - Answer Ms. Acosta received a job as a teacher with a starting salary of $34,000. According to her contract, she will receive a 1.5% increase in her salary every year. How much will Ms. Acosta earn in 7 years? 𝑦=34,000 1+0.015 7 =$3773.47

Partner 2: Partner 2: You are the Sage. You need to verbally explain how to solve the problem step by step. You CANNOT write anything Partner 1: You are the scribe. You CANNOT say anything and can ONLY write down EXACTLY what the Sage tells you to write down.

Example 5 The Work-Out Gym sold 550 memberships in 2001. Since then, the number of memberships sold has increased 3% annually. If this trend continues, predict how many memberships the gym will sell in 2020.

Example 5 - Answer The Work-Out Gym sold 550 memberships in 2001. Since then, the number of memberships sold has increased 3% annually. If this trend continues, predict how many memberships the gym will sell in 2020. 𝑦=5500 1.03 19 =9644 𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝𝑠