Objectives: Today we will … 1.Write and solve exponential growth functions. 2.Graph exponential growth functions. Vocabulary: exponential growth Exponential Growth Functions 8.5

Would You Rather … ?!?! After arguing with your family that you should get a higher allowance your family offers you two allowance options. Either, they will give you $20 each week or they will give you one penny on the first day and double your allowance every day for 31 days. What option would you pick? Back-up your answer with math!

The Solution … 1 $.01 2 $.02 3 $.04 4 $.08 5 $.16 6 $.32 7 $.64 8 $ $ $ $ $ $ $ $ $ $ $ $ $ $10, $20, $41, $83, $167, $335, $671, $1,342, $2,684, $5,368, $10,737,418.24

Real World Exponential Growth Example ential-growth/exponential-models-in-real- world.phphttp:// ential-growth/exponential-models-in-real- world.php

Exponential Growth Functions 8.5 E XPONENTIAL G ROWTH M ODEL A quantity is growing exponentially if it increases by the same percent in each time period. C is the initial amount. t is the time period. (1 + r) is the growth factor, r is the growth rate. Exponential growth always has a growth rate greater than or equal to one. (1 + r) ≥ 1 y = C (1 + r) t Sometimes use P instead of C Note: measure of rate and time MUST be in the same time unit

Example 1 Compound Interest You deposit $1500 in an account that pays 2.3% interest compounded yearly, 1)What was the initial principal ( C ) invested? 2)What is the growth rate ( r )? The growth factor? 3)Using the equation y = C(1+r) t, write the equation that models this situation. Then figure out how much money would you have after 2 years if you didn’t deposit any more money? C or P = $1500 Growth rate (r) is The growth factor is y = $

Example 2 Compound Interest A savings certificate of $1000 pay 6.5% annual interest compounded yearly. First, write the equation that models this situation. Then figure out what is the balance when the certificate matures after 5 years? ≈ $

1.What is the percent increase each year? 2.Write a model for the number of rabbits in any given year. 3.Find the number of rabbits after 5 years. Example 3 Exponential Growth Model A population of 20 rabbits is released into a wild- life region. The population triples each year for 5 years. 200% ≈ 4860 rabbits y =20(1+2.00) t

Exponential Growth Model Graph the growth of the rabbits. Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t y Time (years) Population P = 20 ( 3 ) t Here, the large growth factor of 3 corresponds to a rapid increase Example 4

1.Write a model for the weight during the first 6 week. 2.Find the weight at the end of six weeks. Example 5 Exponential Growth Model A newly hatched channel catfish typically weighs about.3 grams. During the first 6 weeks of life, its growth is approximately exponential, increasing by about 10% a day. y =.3(1+.10) t ≈ 16.4 grams

Example 6 Exponential Growth Model Graph Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t y 0 y =3(1.10) t

1.Write a model for the number of bacteria at any hour. 2.Find the number of bacteria after 8 hours. Example 7 Exponential Growth Model An experiment started with 100 bacteria. They double in number every hour. y =100(1+1.00) t ≈ 25,600 bacteria

pgs #1, 4, 5, 14, 15, 21, 22, 24 Homework