Quiz

1
Quiz

• In this equation, y 0.65(1.3)x , what is base,
  0.65, 1.3, or x ?

2
Write

• 1. What is standard form of exponential function?
• 2. What is growth factor?
• 3. In exponential function, y abx ,
• a) Why is a ? 0?
• b)Why is b ? 1?

3
Review

• 1. What is standard form of exponential function?
• Standard form of exponential function is
• y abx.

4
Review

• 2. What is growth factor?
• The factor which increases the quantity.
• In exponential function y abx,
• a is constant,
• x is a real number, and
• b is growth factor.

5
Review

• 3. In exponential function, y abx ,
• a) Why is a ? 0?
• b) Why is b ? 1?

6
Review

• When a 0, y 0, regardless of the values of b
  and x.
• y abx
• When b 1, bx 1, regardless of the value
  of x.
• y abx

7
What We Will do Today

• First We will review factor.
• Second We will discuss decay.
• Third We will work together to solve some
  problems.
• Finally, we will review what we have learned.

8
Decay

• Before we go on to discuss exponential decay, we
  should pause and discuss some of the terms that
  are frequently used in these problems.

9
Decay

• First, let's talk about the term factor.
• What is a factor?
• Factor is any of the numbers or symbols in
  mathematics that when multiplied together form a
  product.

10
Decay

• Is 2 a factor of 8?
• Yes.
• Why?
• If you multiply 2 with 4, it gives a product of
  8.

11
Decay

• Is 2 a factor of 9?
• No.
• Why not?
• 2 is not a multiple of 9, another words, 2 does
  not gives product of 9, if you multiply with any
  other number.

12
Decay

• Suppose you invested 100, and after a time, your
  investment was worth 300.
• The final value (300) would be three (3) times
  the initial value.
• We would say that your investment had increased
  by a factor of 3.

13
Decay

• On the other hand, if you made a poor investment
• The value decreased from 100 to 25.
• The final value would be a quarter (1/4) of the
  initial value.
• We would say the investment had decreased by a
  factor of 4.

14
Decay

• We learned that, when b gt 1, the function models
  exponential growth, and b is the growth factor.
• But,when 0 lt b lt 1, the function models
  exponential decay,
• and b is the decay factor.

15
Decay

• The standard form of an exponential function is
  y abx ,
• where a is constant, and a cannot be 0.
• b gt 0, and b cannot be equal to 1.

16
Decay

• An exponential function with b gt 0 can be used to
  model either growth or decay, depending of the
  value of b.
• Exponential decay is the decrease in a quantity.

17
Decay

• Same way depreciation is the decline in an items
  value resulting from age and/or wear.
• When an item loses about the same percent of its
  value each year,
• You can use an exponential function to model the
  depreciation or decay.

18
Decay

• You could call decay as negative growth.

19
Group work Solve the following

• A car dealer provides the exponential decay graph
  which shows expected decrease in value of a
  Minivan.

20
Group work

• 1. Estimate the percent decrease in the value of
  the minivan in the first year.

21
Group Work

• The initial value of the minivan is 20,000.
• After one year, according to the graph, the value
  of the van goes down to about 17,000.

22
Group Work

• Initial
  value final value
• Percent decrease ------------------------------
  --
• 
  Initial value
• 20,000 17,000
  3,000
• --------------------
  ------------------ 0.15
• 20,000
  20,000
• The value decrease, or depreciate, 15 in the
  first year.

23
Group Work

• 2. Use y abx, to estimate the value of the
  minivan after 6 years.

24
Group Work

• Let say, x number of year
• y value of the minivan
• The initial value a 20,000
• The decay factor b is (1 0.15) 0.85
• y abx
• 20,000(0.85)x, a 20,000, and b 0.85
• 20,000(0.85)6, substitute 6 for x
• 7542.99
• The minivans value after 6 years will be about
  7500.

25
Group Work

•     3. If the decay factor of 0.85, what will be
  the value of the minivan after 10 years?
• y abx
• 20,000(0.85)10
• 3937.49

26
Group Work

• 4. If the decay factor is 0.65what will be the
  value of minivan after 15 years.
• y abx
• 20,000(0.65)15
• 31.24

27
Group Work

• 5. Use the graphing calculators TRACE feature to
  determine when the minivan will first be worth
  less than 2,000.

28
Group Work

• After about 14 year.

29
Group Work

• 6. A stereo costs 900. Its value is estimated to
  decrease by 12 each year. Write a exponential
  function that models the value of the stereo.

30
Group Work

• Let say, x number of year
• y value of the stereo
• The initial value a 900
• The decay factor b is (1 0.12) 0.88
• y abx
• y 900(0.88)x

31
Summary

• 1. What is decay factor?
• The standard form of an exponential function is
  y abx ,
• when 0 lt b lt 1, the function models exponential
  decay,
• and b is the decay factor.

32
Summary

• 2. How to find depreciation?
• Depreciation is decrease in quantity.
• In exponential function, y abx,
• When 0 lt b lt 1, the function models
  exponential decay.
• Exponential decay is the decrease in a quantity,
  same as depreciation.
•  

33
Summary

• 3. How to find percent decrease?
• Initial value
  final value
• Percent decrease --------------------------
• Initial
  value

34
Homework

• Page 308, 4, 8,
• Page 309, 20, 22