Lesson 6.2 Exponential Equations In this lesson you will ● write exponential equations to represent situations involving a constant multiplier ● change expressions from expanded form to exponential form ● use exponential equations to model exponential growth

Recursive routines are useful unless you need to find a very large term like the 50th term or 60th term. Sequences in Ch 3, for the most part, graphed as lines, so you had to learn how to write a linear equation. Recursive routines with a constant multiplier create a type of increasing or decreasing pattern that we will be able to represent with an equation. We will learn how to represent these recursive routines as equations and we will be able to find any term without knowing the previous term.

A recursive routine that uses a constant multiplier represents a pattern that increases or decreases by a constant ratio or a constant percent. Because exponents are another way of writing repeated multiplication, you can use exponents to model these patterns. In the investigation, you discovered how to calculate the Koch curve at any stage by using this equation: Stage Number y = 27 (4/3)x Total length Constant multiplier Starting length

Equations like this are called Exponential Equations because a variable, in this case x, appears in the exponent. The standard form of an exponential equation is y = a • bx. When you write out a repeated multiplication expression to show each factor, it is written in expanded form. When you show a repeated multiplication expression with an exponent, it is in exponential form and the factor being multiplied is called the base.

Expanded form Exponential form The exponent means there are 3 such factors of 4/3 = 27 (4/3) (4/3) (4/3) 27 (4/3)3 There are three factors of 4/3 The base means you are multiplying factors of 4/3 Example A Write each expression in exponential form. a. (5)(5)(5)(5)(5)(5) b. 3(3)(2)(2)(2)(2)(2)(2)(2)(2)(2) c. the current balance of a saving account that was opened 7 years ago with $200 earning 2.5% interest per year.

Solution to Example A Write each expression in exponential form. b. 3(3)(2)(2)(2)(2)(2)(2)(2)(2)(2) c. the current balance of a saving account that was opened 7 years ago with $200 earning 2.5% interest per year. Solution to Example A Write each expression in exponential form. a. (5)(5)(5)(5)(5)(5) = 56 b. 3(3)(2)(2)(2)(2)(2)(2)(2)(2)(2) = 32• 29 c. the current balance of a saving account that was opened7 years ago with $200 earning 2.5% interest per year. 200 (1 + 0.025)7

Example B: Seth deposits $200 in a savings account Example B: Seth deposits $200 in a savings account. The account pays 5% annual interest. Assuming that he makes no more deposits and no withdrawals, calculate his new balance after 10 years. Answer Expanded form Exponential form New Starting balance $200 Balance After 1 year $200(1+0.05) = $200(1+0.05)1 = $210.00 After 2 years $200(1+0.05)(1+0.05) = $200(1+0.05)2 = $220.50 After 3 years $200(1+0.05)(1+0.05)(1+0.05) = $200(1+0.05)3 = $231.53 After x years $200(1+0.05)(1+0.05)…(1+0.05) = $200(1+0.05)x = You can use the formula to find the (y) in dollars after (x) 10 years. y = $200(1+0.05)x y = $200(1+0.05)10 ≈ 325.78 Amounts that increase by a constant percent, like the savings account in the example, have exponential growth.

Exponential Growth Any constant percent of growth can be modeled by the exponential equation y = A (1 +r )x Where A is the starting value, r is the rate of growth written as a positive decimal or fraction, x is the number of time periods elapsed, and y is the final value.

Examples y = a•bx y = 62.5•(0.2)x ENTER ENTER 1.75 (1.25)x For the table find a and b such that y = a•bx Given the following recursive routine write an exponential equation that will match 1.75 , Ans •(1+.25) x y 1 12.5 2 2.5 4 0.1 5 0.02 y = a•bx 2.5/12.5 = .2 and .02/.1=.2 Current /previous to find b = .2 , To find a, I used the 1st term12.5 /.2 = 62.5 y = 62.5•(0.2)x ENTER ENTER 1.75 (1.25)x

1. Desi bought a car for $16,000. He estimates that the value of his car will decrease by 15% a year. What will the car be worth after 4 years? After 7 years? 2. Six years ago, Dawn’s grandfather gave her a coin collection worth $350. Since then, the value of the collection has increased by 7% per year. How much is the collection worth now? 3. Eight hours ago there were 120 bacteria in a petri dish. Since then the population has increased by 75% each hour. a. How many bacteria are in the population now? b. How many bacteria were in the population 5 hours ago? 8352.10 5129.23 525.26 a. 10,555.67 b. 643.13