Chapter 2

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Chapter 2 Linear and Exponential Functions
2.1 Introducing Linear Models 2.2 Introducing
Exponential Models 2.3 Linear Model Upgrades
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2.1
A linear function models any process that has a
constant rate of change. m
The graph of a linear function is a straight line.
A linear function has the form y f(x) b
mx where f is the name of the function. b is
the starting value or y intercept (f(0)). m is
the constant rate of change or slope.
slope intercept form
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2.1
In summer of 2001, the exchange rate for the
Mexican peso was 9.2.
x(dollar) 0 1 2 3 10
y(peso) 0 9.2 18.4 27.6 92
x 0 to 1 1 to 2 0 to 3 1 to 10
change in x 1 1 3 9
y 0 to 9.2 9.2 to 18.4 0 to 27.6 9.2 to 92
change in y 9.2 9.2 27.6 82.8
m 9.2/1 9.2 9.2/1 9.2 27.6/3 9.2 82.8/9 9.2
CONSTANT RATE OF CHANGE
Mexican peso conversion is a linear function with
respect to US dollar.
4
2.1
In summer of 2001, the exchange rate for the
Mexican peso was 9.2.
pesos
dollars
straight line graph
Mexican peso conversion is a linear function with
respect to US dollar.
5
2.1
In summer of 2001, the exchange rate for the
Mexican peso was 9.2.
p(d) 0.92d
linear formula f(x) b mx starting
value/y-intercept (b) is 0. rate of change/slope
(m) is 0.92.
Mexican peso conversion is a linear function with
respect to US dollar.
6
2.1
Jason decides to purchase a 3000 DJ system that
has a life expectancy of 10 years. He assumes
the value of the equipment will depreciate
linearly by the same amount (300) each year .
x(age) 0 1 2 3 4 5 6 7 8 9 10
y(value) 3000 2700 2400 2100 1800 1500 1200 900 600 300 0
x 0 to 1 1 to 2 0 to 5 3 to 10
change in x 1 1 5 7
y 3000 to 2700 2700 to 2400 3000 to 1500 2100 to 0
change in y -300 -300 -1500 -2100
m -300/1 -300 -300/1 -300 -1500/5 -300 -2100/7 -300
CONSTANT RATE OF CHANGE
Value of DJ system is a linear function with
respect to age.
7
2.1
Jason decides to purchase a 3000 DJ system that
has a life expectancy of 10 years. He assumes
the value of the equipment will depreciate
linearly by the same amount (300) each year .
value(dollars)
age (years)
straight line graph
Value of DJ system is a linear function with
respect to age.
8
2.1
Jason decides to purchase a 3000 DJ system that
has a life expectancy of 10 years. He assumes
the value of the equipment will depreciate
linearly by the same amount (300) each year .
v(t) 3000 - 300t
linear formula f(x) b mx starting
value/y-intercept (b) is 3000 . rate of
change/slope (m) is -300 per year.
Value of DJ system is a linear function with
respect to age.
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2.1
Under America Onlines Unlimited Usage plan, a
member is charged 21.95 per month regardless of
the number of hours spent online. Express the
monthly bill as a function of the number of hours
used in one month.
t(hours) 0 1 2 10 20 100
bill(dollars) 21.95 21.95 21.95 21.95 21.95 21.95
x 0 to 1 1 to 2 2 to 10 1 to 20
change in x 1 1 8 19
y 21.95 to 21.95 21.95 to 21.95 21.95 to 21.95 21.95 to 21.95
change in y 0 0 0 0
m 0/1 0 0/1 0 0/8 0 0/19 0
CONSTANT RATE OF CHANGE
Monthly bill is a linear function with respect to
number of hours used.
10
2.1
Under America Onlines Unlimited Usage plan, a
member is charged 21.95 per month regardless of
the number of hours spent online. Express the
monthly bill as a function of the number of hours
used in one month.
bill (dollars)
time (hours)
STRAIGHT LINE GRAPH
Monthly bill is a linear function with respect to
number of hours used.
11
2.1
Under America Onlines Unlimited Usage plan, a
member is charged 21.95 per month regardless of
the number of hours spent online. Express the
monthly bill as a function of the number of hours
used in one month.
U(t) 21.95
linear formula f(x) b mx starting
value/y-intercept (b) is 21.95 . rate of
change/slope (m) is 0 per hour.
Monthly bill is a linear function of number of
hours spent online.
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2.1
Not all straight line graphs are linear functions.
Consider the equation x 3.
x 3 3 3 3 3
y -4 -1 0 3 5
x 3 to 3 3 to 3 3 to 3 3 to 3
change in x 0 0 0 0
y -4 to 1 -4 to 0 -1 to 0 0 to 5
change in y 5 4 1 5
m 5/0 u 4/0 u 1/0 u 5/0 u
linear formula f(x) b mx
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An exponential function models any process in
which function values change by a fixed ratio or
percentage.
The graph of an exponential function is curvy.
An exponential function has the form y f(x)
c ax where f is the name of the function. c
is the starting value or y intercept (f(0)). a
is the growth factor.
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2.2
Harmful kitchen bacteria can double their numbers
every 20 minutes. A single bacterium on a wet
countertop might in just eight hours, reproduce
to nearly 17 million.
t(20 minute intervals) 0 1 2 3 4 5
P(number of bacteria) 1 2 4 8 16 32
t 0 to 1 1 to 2
change in t 1 1
P 1 to 2 2 to 4
change in P 1 2
m 1/1 1 2/1 2
NO CONSTANT RATE OF CHANGE increasing.
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2.2
Harmful kitchen bacteria can double their numbers
every 20 minutes. A single bacterium on a wet
countertop might in just eight hours, reproduce
to nearly 17 million.
t(20 minute intervals) 0 1 2 3 4 5
P(number of bacteria) 1 2 4 8 16 32
ratio of consecutive output values ratio of consecutive output values
t P(t1)/P(t)
0 P(1)/P(0) 2 / 1 2
1 P(2)/P(1) 4 / 2 2
2 P(3)/P(2) 8 / 4 2
Growth factor is 2 doubling.
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Harmful kitchen bacteria can double their numbers
every 20 minutes. A single bacterium on a wet
countertop might in just eight hours, reproduce
to nearly 17 million.
bacteriapopulation
time (20-minute intervals)
GRAPH IS CONCAVE UP increasing rate of change.
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Harmful kitchen bacteria can double their numbers
every 20 minutes. A single bacterium on a wet
countertop might in just eight hours, reproduce
to nearly 17 million.
P(t) 2t
exponential formula f(x) cax starting
value/y-intercept (c) is 1 bacteria.growth
factor (a) is 2.
Bacteria population is an exponential function of
time.
After 8 hours (24 20-minute time
intervals) P(24) 224 16,777,216 bacteria
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During the late twentieth century, WHO adopted as
one of its goals the elimination of polio
throughout the world. From 1988 to 1996, cases
of polio decreased by roughly 25 annually.
t(years since 1988) 0 1 2 3
P(polio cases) 38,000 38000-.2538000 28500 28500-.2528500 21375 21375-.2521375 16031
t 0 to 1 1 to 2
change in t 1 1
P 38000 to 28500 28500 to 21375
change in P -9500 -7125
m -9500 -7125
NO CONSTANT RATE OF CHANGE increasing.
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During the late twentieth century, WHO adopted as
one of its goals the elimination of polio
throughout the world. From 1988 to 1996, cases
of polio decreased by roughly 25 annually.
t(years since 1988) 0 1 2 3
P(polio cases) 38,000 38000-.2538000 28500 28500-.2528500 21375 21375-.2521375 16031
ratio of consecutive output values ratio of consecutive output values
t P(t1)/P(t)
0 P(1)/P(0) 28500 / 38000 .75
1 P(2)/P(1) 21375 / 28500 .75
2 P(3)/P(2) 16031 / 21375 .7499
growth factor is 0.75 decreasing by 25 means
75 remains
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During the late twentieth century, WHO adopted as
one of its goals the elimination of polio
throughout the world. From 1988 to 1996, cases
of polio decreased by roughly 25 annually.
number of polio cases
years since 1988
GRAPH IS CONCAVE UP increasing rate of change.
21
During the late twentieth century, WHO adopted as
one of its goals the elimination of polio
throughout the world. From 1988 to 1996, cases
of polio decreased by roughly 25 annually.
P(t) 38000(.75)t
exponential formula f(x) cax starting
value/y-intercept (c) is 38000 polio
cases.growth factor (a) is 0.75.
Number of polio cases is an exponential function
of time.
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Chapter 2 Linear and Exponential Functions
HWp81 1-6, 13-18, 21-23 TURN IN 13, 16,
22,