Lesson 6.2: Exponential Equations

1
Lesson 6.2 Exponential Equations

• To explore exponential growth and decay
• To discover the connection between recursive and
  exponential forms of geometric sequences

2
Recursive Routines

• Recursive routines are useful for seeing how a
  sequence develops and for generating the first
  few terms.
• But if youre looking for the 50th term, youll
  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.

3
Recursive Routines

• Recursive routines with a constant multiplier
  create a different pattern. In this lesson
  youll discover the connection between these
  recursive routines and exponents.
• Then with a new type of equation youll be able
  to find any term in a sequence based on a
  constant multiplier without having to find all
  the terms before it.

4
Loosing Area in a Square Fractal

• In this investigation you will look for patterns
  in area of a square fractal.

5
Growth of a Rectangular Fractal

• To create a fractal we will begin with a 27 x 27
  square
• This is stage 0.

6

• To create stage 1 draw two vertical lines and two
  horizontal lines to subdivide the shape into 9
  equal parts. Shade any one of the parts to
  illustrate the square being removed.

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• To create stage 2 draw two vertical lines and two
  horizontal lines in each of the remaining squares
  to subdivide the remaining squares into 9 equal
  parts each. Shade the same one part of each of
  these squares to represent them being removed.

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• To create stage 3 draw two vertical lines and two
  horizontal lines in each of the remaining squares
  to subdivide the square into 9 equal parts.
  Shade the same one part of each of these squares
  to represent the square being removed.

9
Lets collect some data
Stage Number Total Unshaded Area Ratio of this Stages area to the previous Stages area
0
1
2
3

10
Lets collect some data
Stage Number Total Unshaded Area Ratio of this Stages area to the previous Stages area
0
1
2
3

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Use the ratio to predict the area of stage 4
Stage Number Total Unshaded Area Ratio of this Stages area to the previous Stages area
0
1
2
3
4
12
Rewrite each total unshaded area using the
constant multiplier.
Stage Number Total Unshaded Area Ratio of this Stages area to the previous Stages area
0
1
2
3
4
13
If x is the stage number write an expression for
the unshaded area in stage x.
Stage Number Total Unshaded Area
0
1
2
3
4
14
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 area of the rectangular fractal.
Stage Number Total Unshaded Area
0
1
2
3
4
15
Total length
Stage Number
Constant multiplier
Starting length
This type of equation is called an exponential
equation.
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The standard form of an exponential equation is
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Exponential Form
Expanded Form
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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.

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Time Expanded Form Exponential Form New Balance
Starting 200 200
After 1 yr. 200(10.05) 200(10.05)1 210
After 2 yrs. 200(10.05)(10.05) 200(10.05)2 220.50
After 3 yrs. 200(10.05)(10.05)(10.05) 200(10.05)3 231.53
After x yrs. 200(10.05)(10.05)(10.05) 200(10.05)x
After 10 years