Exponential Modeling. Growth and Decay

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1 Exponential Modeling Growth and Decay

2 Identify each as growth or Decay What you should Know

3 y Exponential functions 0<b<1 decay b>1 Growth Review

4 WRITING EXPONENTIAL GROWTH MODELS A quantity is growing exponentially if it increases by the same percent in each time period. EXPONENTIAL GROWTH MODEL a is the initial amount. x is the time period. y = a (1 + r) x (1 + r) is the growth factor, r is the growth rate. The percent of increase is 100r.

5 Finding the Balance in an Account COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? SOLUTION METHOD 1 SOLVE A SIMPLER PROBLEM Find the account balance A 1 after 1 year and multiply by the growth factor to find the balance for each of the following years. The growth rate is 0.08, so the growth factor is = A 1 = 500(1.08) = 540 A 2 = 500(1.08)(1.08) = A 3 = 500(1.08)(1.08)(1.08) = A 6 = 500(1.08) Balance after one year Balance after two years Balance after three years Balance after six years

6 Finding the Balance in an Account COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? SOLUTION METHOD 2 USE A FORMULA Use the exponential growth model to find the account balance A. The growth rate is The initial value is 500. EXPONENTIAL GROWTH MODEL C 500 is the is the initial initial amount. 6 t is is the time period. y = C (1 + r) t A 6 = 500 ( ) (1 + (1 0.08) + r) is is the the growth factor, factor, 0.08 r is the growth rate. rate. The percent of increase is 100r. A 6 = 500(1.08) Balance after 6 years 6

7 Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.

8 Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. a. What is the percent of increase each year? SOLUTION The population triples each year, so the growth factor is r = 3 So, the growth rate r is 2 and the percent of increase each year is 200%. Reminder: percent increase is 100r.

9 Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. b. What is the population after 5 years? Help SOLUTION After 5 years, the population is P = C(1 + r) t Exponential growth model = 20(1 + 2) 5 = Substitute C, r, and t. Simplify. = 4860 Evaluate. There will be about 4860 rabbits after 5 years.

SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through

10 Population GRAPHING EXPONENTIAL GROWTH MODELS A Model with a Large Growth Factor Graph the growth of the rabbit population. SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t P P = 20 ( 3 ) t Here, the large growth factor of 3 corresponds to a rapid increase Time (years)

11 WRITING EXPONENTIAL DECAY MODELS A quantity is decreasing exponentially if it decreases by the same percent in each time period. a is the initial amount. EXPONENTIAL DECAY MODEL x is the time period. y = a (1 r) x (1 r ) is the decay factor, r is the decay rate. The percent of decrease is 100r.

12 Writing an Exponential Decay Model COMPOUND INTEREST From 1982 through 1997, the purchasing power of a dollar decreased by about 3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997? SOLUTION Let y represent the purchasing power and let t = 0 represent the year The initial amount is $1. Use an exponential decay model. y = C (1 r) t = (1)( ) t = t Exponential decay model Substitute 1 for C, for r. Simplify. Because 1997 is 15 years after 1982, substitute 15 for t. y = Substitute 15 for t The purchasing power of a dollar in 1997 compared to 1982 was $0.59.

Purchasing Power (dollars) GRAPHING EXPONENTIAL DECAY MODELS Graphing the Decay of Purchasing

Use the graph to estimate the value of a dollar in ten years.

curve through the points. t 0 1 2 3 4 y 1.00 0.965 0.931 0.899 0.867 0.837 5 6 7 8 9 0.808 0.

13 Purchasing Power (dollars) GRAPHING EXPONENTIAL DECAY MODELS Graphing the Decay of Purchasing Power Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years. Help SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t y Your dollar of today will be worth about 70 cents in ten years. y = t Years From Now

GRAPHING EXPONENTIAL DECAY MODELS CONCEPT SUMMARY EXPONENTIAL GROWTH AND DECAY MODELS EXPONENTIAL GROWTH MODEL y = C (1 + r) t EXPONENTIAL DECAY MODEL y = C (1 r) t An exponential model y = a b t

14 GRAPHING EXPONENTIAL DECAY MODELS CONCEPT SUMMARY EXPONENTIAL GROWTH AND DECAY MODELS EXPONENTIAL GROWTH MODEL y = C (1 + r) t EXPONENTIAL DECAY MODEL y = C (1 r) t An exponential model y = a b t represents exponential growth if b > 1 and exponential decay if 0 < b < 1. (1 + r) is the growth factor, (1 r) is the decay factor, r is the growth C t is the initial time period. amount. (0, C) rate. r is the decay (0, rate. C) 1 + r > 1 0 < 1 r < 1

15 Read and take notes on Methods and Meanings, on page 83 Do the problems: Review and Preview Page 84 #86-91 On your own

Unit 7 Exponential Functions. Name: Period:

Unit 7 Exponential Functions. Name: Period: Unit 7 Exponential Functions Name: Period: 1 AIM: YWBAT evaluate and graph exponential functions. Do Now: Your soccer team wants to practice a drill for a certain amount of time each day. Which plan will