Exponential Functions and Their Graphs/ Compound Interest 2015/16

2 A True Story Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

3 3.1 Exponential Functions and their Graphs Objective: To use exponential functions to solve real life problems. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

4 The exponential function f with base a is defined by f(x) = a x where a > 0, a  1, and x is any real number. For instance, f(x) = 3 x and g(x) = 0.5 x are exponential functions. Definition of Exponential Function

5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The value of f(x) = 3 x when x = 2 is f(2) = 3 2 = The value of f(x) = 3 x when x = –2 is 9 f(–2) = 3 –2 = Example: Exponential Function

6 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The graph of f(x) = a x, a > 1 y x (0, 1) Domain: (– ,  ) Range: (0,  ) Horizontal Asymptote y = 0 Graph of Exponential Function (a > 1) 4 4

7 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The graph of f(x) = a x, a > 1 y x (0, 1) Domain: (– ,  ) Range: (0,  ) Horizontal Asymptote y = 0 Graph of Exponential Function (0 < a < 1) 4 4 -

8 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Sketch the graph of f(x) = 2 x. x xf(x)f(x)(x, f(x)) -2¼(-2, ¼) ½(-1, ½) 01(0, 1) 12(1, 2) 24(2, 4) y 2–2 2 4 Example: Graph f(x) = 2 x

9 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Sketch the graph of g(x) = 2 x – 1. State the domain and range. x y The graph of this function is a vertical translation of the graph of f(x) = 2 x down one unit. f(x) = 2 x y = –1 Domain: (– ,  ) Range: (–1,  ) 2 4 Example: Translation of Graph

10 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Sketch the graph of g(x) = 2 -x. State the domain and range. x y The graph of this function is a reflection the graph of f(x) = 2 x in the y- axis. f(x) = 2 x Domain: (– ,  ) Range: (0,  ) 2 –2 4 Example: Reflection of Graph

11 Example Each of the following graphs is a transformation of What transformation has taken place in each graph? a) b) c) d) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The graph of h is the graph of f shifted one unit left. The graph of g is the graph of f shifted down 2 units. The graph of k is the graph of f reflected in the x-axis. The graph of j is the graph of f reflected in the y-axis.

12 Compound Interest Formulas Interest Compounded Annually: A = Accumulated amount after t years P = principle (invested amount) r = the interest rate in decimal form (5% =.05) t = time in years Interest Compounded n times per year: n = # of times per year interest is compounded (i.e. n = 1 is yearly, n = 12 is monthly, n = 365 is daily) Continuously compounded Interest: e is the natural number Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

13 Example: Invest $1 for 1 year at 100% Consider how n affects the accumulated amount: Yearly n = 1 Bi-annually n = 2 Monthly n = 12 Dailyn = 365 Hourlyn = 8760 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. $2 $2.613 $2.714 $2.25 $2.718

14 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The irrational number e, where e  … is used in applications involving growth and decay. Using techniques of calculus, it can be shown that The number e

15 Example A total of $12000 is invested at 7% interest. Find the balance after 10 years if the interest is compounded: a) quarterly b) continuously Copyright © by Houghton Mifflin Company, Inc. All rights reserved. $24,165 $24,019

16 Student Example One thousand dollars is invested in an account that earns 12% interest compounded monthly. Determine how much the investment is worth after 2 years. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. $1,269.73

17 Student Example The value of a new $500 television decreases 10% per year. Find its value after 5 years. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. $295.24

18 Student Example One hundred dollars is invested at 7.2% interest compounded quarterly. How much money will there be after 6 years? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. $153.44

19 Student Example Let y represent a mass of radioactive strontium, in grams, whose half-life is 28 years. The quantity of strontium present after t years is a) What is the initial mass (when t=0)? b) How much of the initial mass is present after 80 years? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 grams 1.38 grams

20 Student Example Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $58,500 per year and the account pays 4.5% interest. How much interest will her deposit earn after 10 years? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. $3,324.63

21 Exit Ticket A student earned $2500 working during the summer and wants to invest the money in a savings account with 7.5% interest compounded quarterly for 3 years to save for college. How much money will the student have saved for college at the end of 3 years? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. $3,124.29

22 HOMEWORK 3.1 pg odd, odd, 65, 67 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.