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## Lesson 6.2: Exponential Equations

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**Lesson 6.2: Exponential Equations**To explore exponential growth and decay To discover the connection between recursive and exponential forms of geometric sequences**Recursive Routines**• Recursive routines are useful for seeing how a sequence develops and for generating the first few terms. • But if you’re looking for the 50th term, you’ll have to do many calculations to find your answer. • In chapter 3, you found that graphs of the points formed a linear pattern, so you learned to write the equation of a line.**Recursive Routines**• Recursive routines with a constant multiplier create a different pattern. In this lesson you’ll discover the connection between these recursive routines and exponents. • Then with a new type of equation you’ll be able to find any term in a sequence based on a constant multiplier without having to find all the terms before it.**Growth of the Koch Curve**• In this investigation you will look for patterns in the growth of a fractal. Stage 0**Draw Stage 1 figure below the Stage 0 figure. The first**segment is drawn for you on the worksheet. Stage 1 should have four segments. Stage 0 Stage 1**Describe the curve’s recursive rule so that someone can**re-create the curve from your description. Stage 0 Stage 1**Using your recursive rule, determine the length of the Stage**1. Record the total length in the chart for Stage 1. Stage 0 Stage 1**Draw Stage 2 and 3 for the fractal. Again, the first**segment for each stage is drawn for you. Stage 0 Stage 1 Stage 2**Record the total length in the chart.**Stage 0 Stage 1 Stage 2**Find the ratio of the total length at any stage to the total**length at the previous stage. • What is the constant multiplier?**Rewrite the Total Length of Stages 1-3 using the constant**multiplier.**Use your constant multiplier from the previous step to**predict the total length of this fractal at Stages 4 and 5.**How many times do you multiply the original length at Stage**0 by the constant multiplier to get the length at Stage 2? Write an expression that calculates the length of Stage 2.**How many times do you multiply the original length at Stage**0 by the constant multiplier to get the length at Stage 3? Write an expression that calculates the length of Stage 3.**If your expressions in the previous two questions do not use**exponents, rewrite them so that they do.**Use your constant multiplier from the previous step to**predict the total length of this fractal at Stages 5.**If x represents the stage number, and y represents the total**length of this fractal at any stage, write an equation to model the total length of this fractal at any stage.**Create a graph for this equation.**• Check the calculator table to see that it contains the same values as your table. • What does the graph tell you about the growth of the Koch curve?**Total length**Stage Number Constant multiplier Starting length This type of equation is called an exponential equation.**Exponential Form**Expanded Form**Example**• 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. • Determine the constant multiplier. • Write an equation that can be used to calculate the yearly total.