Transcript Document

UNIT 5: Exponential Growth / Decay Formula:
y  ab
a = original amount (y-intercept)
x
b = growth factor (1 ± r)
y = final amount
x = unit of measure (time, bounces, etc.)
Exponential Growth
Exponential Decay
Things to know about… y  ab
• b cannot be negative
b > 1: growth
x
0 < b < 1: decay
• DOMAIN of all exponential functions is:
ALL REAL NUMBERS (no restrictions for x)
• RANGE of exponential functions:
a = positive  Range: y > 0
a = negative  Range: y <0
• Y – INTERCEPT = a
y-values follow multiply or divide patterns
Example 1:Identify a Multiply Pattern to write an equation
3, 6, 12, 24, …
162, 54, 18, …
32, 56, 98, 171.5,…
72, 36, 18, 9, …
4, 12, 36, 108,…
20, 30, 45, 67.5, …
Example 2 Identifying Initial Value, Growth & Decay
EQUATION
y = 2(1.03)x
y = 20(0.85)x
y = 900 (1.27)x
y = 0.75(1.2)x
y = 750(1.15)x
y = 250,000(0.65)x
y = 11,275(1.1)x
y = (1.54)X 1.85
Initial Value
Growth or Decay?
Percent of Change
Simplifying Exponential Expressions
LAWS OF EXPONENTS
• Remember when you multiply terms with same base,
ADD exponents
2 2
5
3
2
5 3
5
4 x3
5
7 x8
11 x  5
5
• When you raise a power to a power, MULTIPLY exponents
(3
x2 3
) 3
3 x6
 4 
 
 5 
5 x 7
8
  4
  
  5 
40 x  56
Practice: Simplify each Expression
1.
4
3.
ab
(7
4
2a  5
x3 2 x
)
2.
4.
a
6
a
3
(a )
3
6
Example 3: Algebraic Solving Exponential Equations
Basic Steps:1]
2]
3]
a)
2
4 n 3
2
3 n 9
FACTOR into common bases
CANCEL common bases
SOLVE equation / inequality
11 x
2 x7
 2
b) 5 21 5 x  5 9  x  2 
 
c)  
 3
 3
Example 3: Not Common Bases
a)
3
2 n 5
 27
b)
4
3 n 2
8
 16 

c) 
 9
4 3 x
 64 
 
 27 
2x
Example 4: Algebraic Solving Exponential Inequalities
2 x  11
6x
3 n 2
n 9
5 x8
2 x 4
a)
7
7
b) 4
4
7
7
c)  
 
 5
5
Example 4: Not Common Bases
a)
5
n 2
 125
b)
32
5 x2
 16
5x
b)
1
 
 3
4 x
1

9
Solving Exponential Equations can be done
with the calculator like rational equations
Calculator Active: [Y=]
Y1 = Left Side of Equation
Y2 = Right Side of Equation
Check your [WINDOW] is large enough
Find INTERSECTION: [2nd] [Trace] [5]
When writing exponents [^] be careful of where you
place your parentheses.
Example 5
x
Word Problems y  ab
1)
The population of rabbits is doubling every 3 months. There
was initially had 15 rabbits. Write an equation to represent this
growth and tell how many rabbits are in the population after 2 years.
2)
The value of a car is depreciating at a rate of 7% each year.
The car was purchased for $36,000 in 2005. What is the value of the
car in 2013?
3)
The number of cell phone sales is expected to increase by 25%
every year. If 10,000 cell phones were sold in 1998, then how many
cell phones were sold in 2005?
4)
The number of mice is growing exponentially each month.
Initially there were only 20 mice. After 2 months, the population of mice
was 180 mice. What is the rate that the mice are growing at? How many
mice would you expect at the end of the year (12 months)?
5)
A bacteria colony is growing exponentially each day.
There was initially had 100 bacteria and after 3 days
it had 800. Write an equation to represent this growth,
and tell how many bacteria after 10 days.
6)
A towns population is growing exponentially. In 2000,
the population was 10,000. By 2006 it had risen to 29,860.
Let x = 0 represent 2000. Write an equation to represent
the growth, and predict the population in 2010.
7)
The increase in gas price is modeled by the equation
y = 2.17(1.023)x, where x is years since 2004. When
would we expect gas prices to be $3.25
8)
The decrease in a computer’s value is modeled by the equation
y = 4000(.87)x, where x is years since 2007. When
would we expect the computer to be worth half it’s value?
9)
y = 10,000(1. 08)x, where x is months, models the
profit earned by a company. When will the company be able to
triple its profit?
10)
y = 40,000(.75)x, where x is years, models the value of a
car. When will the car be worth only $10,000?