5-Minute Check on Activity 5-4

1
5-Minute Check on Activity 5-4

• Determine the domain, range and horizontal
  asymptotes for
• y 4x domain range
  HA
• y (1/2)x domain range
  HA
• Compare the transformations between the two
  functions, f and g
• f(x) 4x and g(x) 4x 3
• f(x) 4x and g(x) 4x 2
• Determine the growth rate and the growth factor
  for y 0.3(1.27)x
• Determine the decay rate and the decay factor for
  y 6(0.87)x

All real
gt 0
y 0
All real
gt 0
y 0
shifts function up by 3
shifts function left by 2
growth factor 1.27 growth rate 0.27 or
27
decay factor 0.87 decay rate -0.13 or
-13
Click the mouse button or press the Space Bar to
display the answers.
2
Activity 5 - 5

• Cellular Phones

3
Objectives

• Determine the growth and decay factor for an
  exponential function represented by a table of
  values or an equation
• Graph exponential functions defined by y abx,
  where a ? 0, b gt 0 and b ? 1
• Identify the meaning of a in y abx as it
  relates to a practical situation
• Determine the doubling and halving time

4
Vocabulary

• Doubling time the time required for the amount
  to double
• Halving time also known as half-life, the time
  required for the amount to decrease by one-half

5
Activity

• During a meeting, you hear the familiar ring of a
  cell phone. Without hesitation, several of your
  friends reach into their jacket pockets, brief
  cases and purses to receive the anticipated call.
  Although sometimes annoying, cell phones have
  become part of our way of life. The following
  table shows the rapid increase in the number of
  cell phone users in the late 1990s.

Year Cell Phones ( in millions)
1996 44.248
1997 55.312
1998 69.14
1999 86.425
2000 108.031
Is this a linear function? Why or why not?
No
rate of change is not constant
6
Activity cont

• Calculate the missing pieces in the table below

Year Cell Phones (in millions) Rate of Change Ratio between Years
1996 44.248
1997 55.312
1998 69.14
1999 86.425
2000 108.031
0 1.25
11.064 1.25
13.828 1.25
17.285 1.25
21.606 1.25
Is the rate of change (slope) the same? Is the
ratio between consecutive years the same?
No
Yes
7
Activity cont

• Does the relationship in the table represent an
  exponential function?
• What is the growth factor?
• Set up an equation, N abt, where N represents
  the number of cell phones in millions and t
  represents the number of years since 1996
• What is the practical domain of the function N?

Yes
1.25
N 44.248 (1.25)t
domain, t 0 (and 5 from table)
8
Exponential Functions

• Exponential Functions of the form y abx, where
  b is gt 0 and b ? 1
• a is called the initial value, y-intercept (0, a)
  at x 0
• Exponential functions have successive ratios that
  are constant
• The constant ratio is a growth factor, if
  y-values are increasing (b gt 1)
• The constant ratio is a decay factor, if y-values
  are decreasing ( 0 lt b lt 1)
• Doubling time set by growth factor
• Half-life is set by the decay factor

9
Exponential Function Identification

• Identify the y-intercept, growth or decay factor,
  and whether the function is increasing or
  decreasing
• f(x) 5(2)x
• g(x) ¾(0.8)x
• h(x) ½ (5/6)x
• f(t) 3(4/3)x

y-int 5, gf 2, increasing
y-int 3/4 df 0.8, decreasing
y-int 1/2 df 5/6, decreasing
y-int 3, gf 4/3, increasing
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Doubling and Halving Times

• Doubling time time it takes for y-values to
  double. It is determined by the growth rate and
  is the same for all y-values
• Half-life time it takes for y-values to decay
  by one-half. It is determined by the decay rate
  and is the same for all y-values
• To find using your calculator Let Y1
  exponential function and Y2 doubled or halved
  amount. Graph them and use 2nd TRACE to find
  Intersection

11
Exponential Growth Example

• An investment accounts balance, B(t), in
  dollars, is defined by B(t) 5500(1.12)t, where
  t is the number of years.
• What was the initial investment?
• What is the interest rate on the account?
• When will the investment double in value?
• When will the investment quadruple in value?

initial investment 5,500
interest rate 12
11000 5500(1.12)t solve graphically t 6.12
years
22000 5500(1.12)t solve graphically t
12.23 years
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Exponential Decay Example

• Chocolate chip cookie freshness decays over time
  due to exposure to air. If the cookie freshness
  is defined by f(t) (0.8)t, find the following
  information.
• What was the initial cookie freshness?
• What is the decay rate on the cookies?
• When will the cookies freshness be halved?

initial cookie freshness was 1
decay rate 1 0.8 0.2 or 20
0.5 1(0.8)t solve graphically t 3.11 days
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Summary and Homework

• Summary
• Functions defined by y abx, where a is the
  initial value and b is the growth or decay factor
  are exponential functions
• Y-intercept is (0, a)
• Growth factor, b gt 1, y-values are increasing
• Decay factor, 0 lt b lt 1, y-values are decreasing
• Doubling time is the time for the y-value to
  double
• Half-life is the time for the y-value to be
  halved
• Homework
• pg 576 580 problems 2, 3, 6, 7