AII, 12.0: STUDENTS KNOW THE LAWS OF FRACTIONAL EXPONENTS, UNDERSTAND EXPONENTIAL FUNCTIONS, AND USE THESE FUNCTIONS IN PROBLEMS INVOLVING EXPONENTIAL GROWTH AND DECAY. IF YOU HAVE A CALCULATOR, USE IT! Modeling with Exponential Functions

Objective Key Words 1. Write models for exponential growth and decay 2. Compound Interest 3. The Natural Base e 4. EC: Write an exponential function Modeling with Exponential Functions

Simplify Greater than 1 or Less than 1 Greater than 1 or Less than 1 Prerequisite Check Can you change percentages to decimals? Yes or No

Percent to Decimals

Exponential Growth Exponential Decay Teacher Input for 1: Write Models

Example 1 Write and Use an Exponential Growth Model Sales Figures One year, a clothing company had $1.5 million in sales. In later years, sales y (in millions of dollars) increased by about 25% each year. a.Write an exponential growth model that represents the sales after t years. b. Use the model to predict the sales after 8 years. SOLUTION a. = ya ()t)t r+1 Write exponential growth model. = 1.5 ()t)t Substitute 1.5 for a and 0.25 for r. = 1.5 ( Simplify. )t)t 1.25

Example 1 Write and Use an Exponential Growth Model ANSWER The model is = 1.5 ()t.)t y b. To predict the sales after 8 years, substitute 8 for t. y = 1.5 ()8) ≈ 8.9 ANSWER The sales after 8 years will be about $8.9 million.

You Try!!! 1.Redo Example 1 using a sales increase of 10% each year. Write and Use an Exponential Growth Model ANSWER = 1.5 ()t)t 1.1 y a. b. $3.2 million

Example 2 Write and Use an Exponential Decay Model Computers You buy a new computer for $1500. The value y (in dollars) of the computer decreases by 40% each year. SOLUTION a. Let t be the number of years since you bought the computer. a.Write an exponential decay model that represents the value of the computer. b. Use the model from part (a) to estimate the value after 3 years. ay = ()t)t 1 – r Write exponential decay model.

Example 2 Write and Use an Exponential Decay Model 1500 = ()t)t 1 – 0.4 Substitute 1500 for a and 0.4 for r = ( Simplify. )t)t 0.6 ANSWER The model is 1500y = ( )t.)t. 0.6 b. To estimate the value after 3 years, substitute 3 for t = ()3)3 0.6 y324 = ANSWER The value of the computer after 3 years is $324.

You Try!!! 2. Redo Example 2 using a new computer cost of $1200 and a value decrease of 35% each year. Write and Use an Exponential Decay Model ANSWER a. 1200y = ()t)t 0.65 b. $329.55

Teacher input for 2: Compound Interest An initial principal P is deposited in an account that pays interest at an annual rate r (expressed as a decimal), compounded n times per year.

Example 3 Find the Balance in an Account Finance You deposit $2000 in an account that pays 2% annual interest. Find the balance after 10 years if the interest is compounded quarterly. Write compound interest formula. = A nt n r SOLUTION 1+ P Substitute 2000 for P, 0.02 for r, 4 for n, and 10 for t. = Simplify. = () 40 Use a calculator ≈

Example 3 Find the Balance in an Account ANSWER The balance after 10 years is about $ Using the graph, you can see that after 10 years, the balance is between $2400 and $2500. CHECKYou can check the solution by graphing the function A = ()4t.)4t.

You Try!!! 6.You deposit $1500 in an account that pays 2% annual interest. Find the balance after 6 years if the interest is compounded monthly. Write and Use Exponential Functions ANSWER $

Teacher input 3: Natural Base e

Example 4 Find the Balance in an Account Finance You deposit $500 in an account that pays 4% annual interest compounded continuously. a. Find the balance after one year. b. Graph the continuously compounded interest model. c.Use the graph to estimate how long it will take your money to grow to $700. Substitute 500 for P, 0.04 for r, and 1 for t. = 500e SOLUTION Write continuously compounded interest formula. = APe rtPe rt a.

Example 4 Find the Balance in an Account Simplify. = 500e 0.04 Use a calculator ≈ ANSWER The balance after one year is about $ c.Use the Trace feature to determine when the balance is $700. This happens when x is about 8.4. So, it will take about 8.4 years for your money to grow to $700. b.The graph of the model is shown at the right. = y500e 0.04x

You Try!!! 7. You deposit $1000 in an account that pays 3% annual interest compounded continuously. Find the balance after 1 year, 3 years, and 5 years. Find the Balance in an Account ANSWER $ , $ , $

Summary Assignment Pg 430 #(14-25,29,30 ALL) EC Pg 431 #(33-43 ODD) Problems not finished in class are left as homework Conclusions

JUST AS TWO POINTS DETERMINE A LINE, TWO POINTS ALSO DETERMINE AN EXPONENTIAL CURVE Write an Exponential Function

Example 5 Write an Exponential Function = yab x 1, 6 () 2, 18 () SOLUTION Substitute the coordinates of the points into to obtain two equations. = yab x Substitute 6 for y and 1 for x, because (1, 6) is on the graph. = 6ab 1 Substitute 18 for y and 2 for x, because (2, 18) is on the graph. = 18ab 2 Solve the first equation for a to get. Then substitute into the second equation. = a b 6 Write a function of the form whose graph passes through and.

Example 5 Write an Exponential Function Substitute for a. = 18b 2b 2 b 6 b 6 Quotient of powers property = 186b6b Divide each side by 6. = 3b ANSWER Using, you find that Because and, is the exponential function whose graph passes through and = b3 = a b 6 = = = a2 = b3 = y3x3x 2 1, 6 () 2, 18 ().

You Try!!! 3. Write an exponential function of the form whose graph passes through the given points. Write and Use Exponential Functions = yab x 2, 16 (),), 3, 64 () ANSWER = y4x4x 4. 2, 3 (),), 4, 12 () ANSWER = y2x2x , 3 (),), 3, 108 () ANSWER = y6x6x 2 1

Practice Work Assignment Complete in class:  Pg459 #(ALL) Ask others for help For teacher help you must provide 3 names of those you asked All 4 of those unable to help will be given help Pg456 #(13-16 ALL) Problems not finished in class are left as homework Conclusion