Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 4 Section 2 1 Exponential Functions

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1 Bob Brown, CCBC Esse Math 163 College Algebra, Chapter 4 Section 2 1 Eponential Functions Motivating Eample Suppose that, on his 18 th birthday, Biff deposits $10,000 into an account that earns 6% annual interest. He does not invest or save any other money, and he retrieves his investment 50 years later on his 68 th birthday. Biff s twin brother, Spudd, does not start saving money until he is 63, and he deposits not $10,000 but $30,000 each year for five years into an account that also pays 6% annual interest. Which brother will have more money in his account on his 68 th birthday? Eponential Functions Def.: An eponential function is a function of the form, where or. It is given the name eponential because the independent variable is in the eponent s place. Note that the restrictions on the possible values of the base a make it so that the domain of f ( ) b is the set of all real numbers. Eercise 1: Graph the function f 2 ( ) by filling in the following table. y -1 y = 2-1 = = = 0.5 Plot the point (-1, 0.5). 0 y = 2 0 = Plot the point (0, ). 1 y = Plot the point 2 y = Plot the point 3 y = Plot the point

Bob Brown, CCBC Esse Math 163 College Algebra, Chapter 4 Section 2 2 Eponential Functions Eercise 2: Graph the function 1 f ( ) using a graphing calculator.

2 Bob Brown, CCBC Esse Math 163 College Algebra, Chapter 4 Section 2 2 Eponential Functions Eercise 2: Graph the function 1 f ( ) using a graphing calculator. 3 Properties of the Eponential Function f ( ) b, b > 1 f ( ) b, 0 < b < 1 1. The domain is 1. The domain is 2. The range is 2. The range is 3. As, y 3. As, y f () As, y As, y The horizontal asymptote is The horizontal asymptote is 4. The function is 4. The function is 5. The function is 5. The function is

3 Bob Brown, CCBC Esse Math 163 College Algebra, Chapter 4 Section 2 3 Eponential Functions Eercise 3: Use transformations to sketch the graph of f ( ) Determine the domain, range, and horizontal asymptote with each transformation. Start with the basic eponential function f 3 ( ). Domain: Range: 0 y Horiz. Asymptote: y = 0 Net, we recognize that 3 2 is a shift of 3 two units to the. Domain: Range: Horiz. Asymptote: Finally, we recognize that is a shift of 3 2 five units. Domain: Range: Horiz. Asymptote:

4 Bob Brown, CCBC Esse Math 163 College Algebra, Chapter 4 Section 2 4 Eponential Functions Financial Application: Interest Interest is money paid for the use of borrowed money. The total amount borrowed (whether by an individual from a bank in the form of a loan or by a bank from an individual in the form of a savings account or a certificate of deposit) is called the principal. The interest rate, epressed as a percent, is the amount charged for the use of the principal for a given period of time. Simple Interest Simple interest is interest calculated on the amount of the original principal alone. Eercise 4: How much interest accumulates in five years on a principal of $10,000 that pays 6% simple annual interest? How much is the total investment worth at the end of the five years? The amount of interest in one year is.06 $10,000 =. 6% of $10,000 In general, the amount of interest in one year would be r P, usually written Pr. Amount of Interest for the Year Current (accumulating) Value of Investment Year 1 Year 2 Year 3 Year 4 Year 5 In general, since the amount of interest for each year is Pr, the total interest would be Pr Pr Pr... Pr = =. t times Simple Interest Formula If P dollars is borrowed for t years at an annual interest rate r (epressed as a decimal), then the interest is given by I = Prt. The total value of the simple interest investment after t years is P + I = + =

5 Bob Brown, CCBC Esse Math 163 College Algebra, Chapter 4 Section 2 5 Eponential Functions Compound Interest Compound interest is interest calculated on the current value of the investment, including on all interest previously earned. Eercise 5: How much is an investment worth in five years on a principal of $10,000 that pays 6% interest compound annually? How much interest was earned during that time? Amount of Interest for the Year Year 1 Year 2 Current Value of Investment So, we can deduce that after five years, the total value of the investment is The amount of interest earned during that time is Eercise 6: How much is an investment worth in five years on a principal of $10,000 that pays 6% interest compound quarterly? How much interest was earned during that time? Note: If interest is compounded quarterly, that means that the interest is calculated and added into the account. However, the rate for each of the quarterly compounding periods is one-fourth of the stated amount. Quarterly rate: One-fourth of 6% = Number of compounding periods: (4 times per year) (5 years) = After 5 years, the total value of the investment is The amount of interest earned during that time is

6 Bob Brown, CCBC Esse Math 163 College Algebra, Chapter 4 Section 2 6 Eponential Functions Compound Interest Formula The amount A after t years due to a principal P invested at an annual interest rate r compounded n times per year is Typically, A is called the and P is called the Eercise 7: How much is an investment worth in five years on a principal of $10,000 that pays 6% interest compound monthly? How much interest was earned during that time? Eercise 8: How much is an investment worth in five years on a principal of $10,000 that pays 6% interest compound daily? How much interest was earned during that time? Continuous Compounding The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is Eercise 9: How much is an investment worth in five years on a principal of $10,000 that pays 6% interest compound continuously? How much interest was earned during that time?

7 Bob Brown, CCBC Esse Math 163 College Algebra, Chapter 4 Section 2 7 Eponential Functions Eercise 10: Solve the problem in the Motivating Eample from page 1. Biff: $10,000 invested at 6% interest compounded annually for 50 years. Spudd: $30,000 added into the account for each of five years at 6% interest compounded annually.

MA Notes, Lesson 19 Textbook (calculus part) Section 2.4 Exponential Functions

MA Notes, Lesson 19 Textbook (calculus part) Section 2.4 Exponential Functions MA 590 Notes, Lesson 9 Tetbook (calculus part) Section.4 Eponential Functions In an eponential function, the variable is in the eponent and the base is a positive constant (other than the number ). Eponential