Section 9BSection 9BLinear ModelingLinear Modeling
Pages 542-553Pages 542-553
Linear FunctionsLinear Functions
A A Linear FunctionLinear Function changes by the changes by the same absolute amountsame absolute amount for each unit for each unit of change in the input (independent of change in the input (independent variable). variable).
A A Linear FunctionLinear Function has a constant has a constant rate of changerate of change..
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Examples:Straightown population as a function of time.
Postage cost as a function of weight.
Pineapple demand as a function of price.
First Class Mail – a linear functionFirst Class Mail – a linear function Weight Postage
cost
1 oz1 oz $0.37$0.37
2 oz2 oz $0.60$0.60
3 oz3 oz $0.83$0.83
4 oz4 oz $1.06 $1.06
5 oz5 oz $1.29$1.29
6 oz6 oz $1.52 $1.52
7 oz7 oz $1.75$1.75
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First Class Mail – a linear functionFirst Class Mail – a linear function Weight Postage
costDifferenc
e
1 oz1 oz $0.37$0.37
2 oz2 oz $0.60$0.60
3 oz3 oz $0.83$0.83
4 oz4 oz $1.06 $1.06
5 oz5 oz $1.29$1.29
6 oz6 oz $1.52 $1.52
7 oz7 oz $1.75$1.75
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First Class Mail – a linear functionFirst Class Mail – a linear function Weight Postage
costDifferenc
e
1 oz1 oz $0.37$0.37
2 oz2 oz $0.60$0.60 $0.23$0.23
3 oz3 oz $0.83$0.83
4 oz4 oz $1.06 $1.06
5 oz5 oz $1.29$1.29
6 oz6 oz $1.52 $1.52
7 oz7 oz $1.75$1.75
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First Class Mail – a linear functionFirst Class Mail – a linear function Weight Postage
costDifferenc
e
1 oz1 oz $0.37$0.37
2 oz2 oz $0.60$0.60 $0.23$0.23
3 oz3 oz $0.83$0.83 $0.23$0.23
4 oz4 oz $1.06 $1.06
5 oz5 oz $1.29$1.29
6 oz6 oz $1.52 $1.52
7 oz7 oz $1.75$1.75
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First Class Mail – a linear functionFirst Class Mail – a linear function Weight Postage
costDifferenc
e
1 oz1 oz $0.37$0.37
2 oz2 oz $0.60$0.60 $0.23$0.23
3 oz3 oz $0.83$0.83 $0.23$0.23
4 oz4 oz $1.06 $1.06 $0.23$0.23
5 oz5 oz $1.29$1.29
6 oz6 oz $1.52 $1.52
7 oz7 oz $1.75$1.75
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First Class Mail – a linear functionFirst Class Mail – a linear function Weight Postage
costDifferenc
e
1 oz1 oz $0.37$0.37
2 oz2 oz $0.60$0.60 $0.23$0.23
3 oz3 oz $0.83$0.83 $0.23$0.23
4 oz4 oz $1.06 $1.06 $0.23$0.23
5 oz5 oz $1.29$1.29 $0.23$0.23
6 oz6 oz $1.52 $1.52
7 oz7 oz $1.75$1.75
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First Class Mail – a linear functionFirst Class Mail – a linear function Weight Postage
costDifferenc
e
1 oz1 oz $0.37$0.37
2 oz2 oz $0.60$0.60 $0.23$0.23
3 oz3 oz $0.83$0.83 $0.23$0.23
4 oz4 oz $1.06 $1.06 $0.23$0.23
5 oz5 oz $1.29$1.29 $0.23$0.23
6 oz6 oz $1.52 $1.52 $0.23$0.23
7 oz7 oz $1.75$1.75
9-B
First Class Mail – a linear functionFirst Class Mail – a linear function Weight Postage
costDifferenc
e
1 oz1 oz $0.37$0.37
2 oz2 oz $0.60$0.60 $0.23$0.23
3 oz3 oz $0.83$0.83 $0.23$0.23
4 oz4 oz $1.06 $1.06 $0.23$0.23
5 oz5 oz $1.29$1.29 $0.23$0.23
6 oz6 oz $1.52 $1.52 $0.23$0.23
7 oz7 oz $1.75$1.75 $0.23$0.23
9-B
First Class PostageFirst Class Postage9-B
First Class Postage Cost as a function of Weight
$0.00
$0.50
$1.00
$1.50
$2.00
0 1 2 3 4 5 6 7 8
Weight (oz)
Po
stag
e
First class postage – a First class postage – a linear functionlinear function
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$0.23$0.23
1rateof change per ounce
oz
First class postage – a First class postage – a linear functionlinear function
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$0.46$0.23
2rateof change per ounce
oz
First class postage – a First class postage – a linear functionlinear function
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$1.15$0.23
5rateof change per ounce
oz
change in variable
change in
depen
ind
den
variablependent
t
erateof change
2
2 1
1(
( )
)yrateof ch
ya
xn
xge
We define ‘rate of change’ We define ‘rate of change’ of a linear function by:of a linear function by:
where (x1,y1) and (x2,y2) are any two ordered pairs of the function.
Slope = rate of changeSlope = rate of change9-B
Linear FunctionsLinear FunctionsA A linear functionlinear function has a has a constant rate of constant rate of
changechange and a and a straight line graphstraight line graph..
The The rate of change = slope of the rate of change = slope of the graphgraph..The greater the rate of change, the The greater the rate of change, the steeper the slope.steeper the slope.
positive slope positive slope negative negative slopeslope
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rise
runslope
Example: Price-Demand Example: Price-Demand FunctionFunction
A linear function is used to describe A linear function is used to describe how the demand for pineapples varies how the demand for pineapples varies with the price.with the price.
($2, 80 pineapples) and ($5, 50 ($2, 80 pineapples) and ($5, 50 pineapples).pineapples).
Find the rate of change (slope) for this Find the rate of change (slope) for this function and then graph the function.function and then graph the function.
independent variable = priceindependent variable = price
dependent variable = demand for dependent variable = demand for pineapplespineapples
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Example: Price-Demand Example: Price-Demand FunctionFunction
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change in demand
change in priceslope
($2, 80 pineapples) and ($5, 50 pineapples)($2, 80 pineapples) and ($5, 50 pineapples)
80 50
$2 $5
pineapples pineapples
30
$3
pineapples
10 /pineapple dollar
50 80
$5 $2
pineapples pineapplesor
($2, 80 pineapples) and ($5, 50 pineapples).($2, 80 pineapples) and ($5, 50 pineapples).
To graph a linear function you need 2 To graph a linear function you need 2 things:things:
• two pointstwo points or or• slope and one pointslope and one point
Example: Price-Demand Example: Price-Demand FunctionFunction
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change in demand10 $
change in priceslope pineapples per
Example: Price-Demand Example: Price-Demand FunctionFunction
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($2, 80 pineapples) and ($5, 50 pineapples).($2, 80 pineapples) and ($5, 50 pineapples).
Demand for Pineapples as a function of Price
0
20
40
60
80
100
$- $2.00 $4.00 $6.00 $8.00 $10.00
Price
Dem
and
Example: Price-Demand Example: Price-Demand FunctionFunction
9-B
($2, 80 pineapples) and ($5, 50 pineapples).($2, 80 pineapples) and ($5, 50 pineapples).
Demand for Pineapples as a function of Price
0
20
40
60
80
100
120
$- $2.00 $4.00 $6.00 $8.00 $10.00
Price
Dem
and
General Equation for a General Equation for a Linear FunctionLinear Function
dependent = initial value + (slope)×independent yy = initial value + (slope)×xx(Initial value occurs when the independent variable =
0.)
y y = = mmxx + + b b or or
y y = b + mx
m = m = slope slope
bb = = yy-intercept -intercept
(The line goes through the point (0,b).)(The line goes through the point (0,b).)
9-B
Example:Example:dep. variable = initial value + (slope)× indep. dep. variable = initial value + (slope)× indep.
variablevariable
slope = -10 pineapples/$slope = -10 pineapples/$ initial value = 100 pineapplesinitial value = 100 pineapples
Demand = 100 - 10×(price)Demand = 100 - 10×(price)DD = 100 – 10 = 100 – 10pp
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Demand for Pineapples as a function of Price
0
20
40
60
80
100
120
$- $2.00 $4.00 $6.00 $8.00 $10.00
Price
Dem
and
Example:Example:
Demand = 100 - 10×(price)Demand = 100 - 10×(price)DD = 100 – 10 = 100 – 10ppCheck: $2: 100 - 10×2 = 80 pineapplesCheck: $2: 100 - 10×2 = 80 pineapples
$5: 100 - 10×5 = 50 pineapples$5: 100 - 10×5 = 50 pineapples
9-B
Demand for Pineapples as a function of Price
0
20
40
60
80
100
120
$- $2.00 $4.00 $6.00 $8.00 $10.00
Price
Dem
and
t P=f(t)
00 f(0)=10,00f(0)=10,0000
55 f(5)=12,50f(5)=12,5000
1010 f(10)=15,0f(10)=15,00000
1515 f(15)=17,5f(15)=17,50000
2020 f(20)=20,0f(20)=20,00000
4040 f(40)=30,0f(40)=30,00000
Growth of Straightown
20, 20000
5, 12500
0, 10000
40, 30000
15, 17500
10, 15000
0
5000
10000
15000
20000
25000
30000
35000
0 10 20 30 40 50
years
po
pu
lati
on
Data Table
Graph
old example: The initial population of Straightown is 10, 000 and increases by 500 people per year.
t P=f(t)
00 10,00010,000
55 12,50012,500
1010 15,00015,000
1515 17,50017,500
2020 20,00020,000
4040 30,00030,000
old example: The initial population of Straightown is 10, 000 and increases by 500 people per year.
15,000-10,000
10-0slope
12,500-10,000
5-0rateof change
20,000-12,500
20-5rateof change
= 500
= 500
= 500
Rate of change (slope) is ALWAYS 500 (people per year).
Initial population is 10,000 (people).
Linear Function: Population = 10,000 + 500×(year)
Example – First class postageExample – First class postage
Weight Postage cost
1 oz1 oz $0.37$0.37
2 oz2 oz $0.60$0.60
3 oz3 oz $0.83$0.83
4 oz4 oz $1.06 $1.06
5 oz5 oz $1.29$1.29
6 oz6 oz $1.52 $1.52
7 oz7 oz $1.75$1.75
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Slope = Slope = $.23/ounce$.23/ounce
initial value = initial value = $0.14$0.14
Example: First Class PostageExample: First Class Postage
Slope = $.23/ounceSlope = $.23/ounce
initial value = $0.14initial value = $0.14
Postage = $0.14 + $0.23×(weight)Postage = $0.14 + $0.23×(weight)
PP = $0.14+ $0.23 = $0.14+ $0.23ww
Check: 1 ounce: $0.14+ $0.23×1 = $0.37Check: 1 ounce: $0.14+ $0.23×1 = $0.37
6 ounces: $0.14 + $0.23×6 = 6 ounces: $0.14 + $0.23×6 = $1.52$1.52
9-B
First class postage as a function of weight
$-
$0.50
$1.00
$1.50
$2.00
0 1 2 3 4 5 6 7 8
Weight (oz)
Po
stag
e
Example:Example:The world record time in the 100-meter butterfly The world record time in the 100-meter butterfly
was 53.0 seconds in 1988. Assume that the was 53.0 seconds in 1988. Assume that the record record fallsfalls at a constant rate of at a constant rate of 0.05 seconds 0.05 seconds per yearper year. What does the model predict for the . What does the model predict for the record in 2010?record in 2010?
dependent variable dependent variable = world record time (R)= world record time (R)independent variableindependent variable is time, is time, tt (years) after (years) after
1988.1988.SlopeSlope = 0.05 seconds; = 0.05 seconds; initial valueinitial value = 53.0 = 53.0
seconds;seconds;Record time = 53.0 – 0.05×(t years after 1988)
R = 53 – 0.05tRecord time in 2010 = 53 - .05×(22) = 51.9 Record time in 2010 = 53 - .05×(22) = 51.9
secondsseconds
9-B
Example:Example:9-B
Suppose you were 20 inches long at birth Suppose you were 20 inches long at birth and 4 ft tall on your tenth birthday. and 4 ft tall on your tenth birthday. Create a Create a linear equationlinear equation that describes that describes how your height varies with age.how your height varies with age.
independent variable = age (years)independent variable = age (years)
dependent variable = height (inches)dependent variable = height (inches)
Two points: (0, 20) (10, 48)Two points: (0, 20) (10, 48)
Initial value = 20 inchesInitial value = 20 inches
Height = 20 + 2.8t t = years
48 202.8 /
10 0slope in yr
Example:Example:9-B
““Fines for Certain PrePayable Violations” – Fines for Certain PrePayable Violations” – Speeding other than residence zone, highway Speeding other than residence zone, highway work zone and school crosswalk: work zone and school crosswalk: $5.00 per $5.00 per MPH over speed limitMPH over speed limit
plus processing fee ($51.00) and local fees plus processing fee ($51.00) and local fees ($5.00)($5.00)
independent variable = miles over speed limitindependent variable = miles over speed limit
dependent variable = fine ($)dependent variable = fine ($)
Initial valueInitial value = $56.00 = $56.00 SlopeSlope = $5.00 = $5.00
Fine = $56 + $5(your speed-speed limit)
Example:Example:9-B
Mrs. M. was given a ticket for doing 52 Mrs. M. was given a ticket for doing 52 mph in a zone where the speed limit mph in a zone where the speed limit was 35 mph. How much was her fine?was 35 mph. How much was her fine?
Fine = $55 + $5(her speed-35)
Fine = $56 + $5(52-35) = $56 + $5(17)
= $141
Example:Example:9-B
““Fines for Certain PrePayable Violations” – Fines for Certain PrePayable Violations” – Speeding in a Speeding in a residence zoneresidence zone: $200 plus : $200 plus $7.00 per MPH over speed limit (25 mph), $7.00 per MPH over speed limit (25 mph), plus processing fee ($51.00) and local fees plus processing fee ($51.00) and local fees ($5.00)($5.00)
independent variable = miles over speed limitindependent variable = miles over speed limit
dependent variable = fine ($)dependent variable = fine ($)
Initial valueInitial value = $256.00 = $256.00 SlopeSlope = $7.00 = $7.00
Fine = $256 + $7(your speed-25)
Example:Example:9-B
The Psychology Club plans to pay a visitor $75 to The Psychology Club plans to pay a visitor $75 to speak at a fundraiser. Tickets will be sold for $2 speak at a fundraiser. Tickets will be sold for $2 apiece. Find a linear equation that gives the apiece. Find a linear equation that gives the profit/loss for the event as it varies with the profit/loss for the event as it varies with the number of tickets sold. number of tickets sold.
independent variable = number of tickets soldindependent variable = number of tickets sold
dependent variable = profit/loss ($)dependent variable = profit/loss ($)
(0, -$75) (0, -$75) slopeslope = +$2 (= rate of change in = +$2 (= rate of change in ticket price)ticket price)
Profit = -$75 +2×(number of tickets)Profit = -$75 +2×(number of tickets)
P = -$75 +2P = -$75 +2nn
Example:Example:9-B
How many people must attend for How many people must attend for the club to break even?the club to break even?
P = -$75 +2n
0 = -$75 + 20 = -$75 + 2nn
$75 = 2$75 = 2nn
37.5 = 37.5 = nn
Can’t sell half a ticket -- so we’ll Can’t sell half a ticket -- so we’ll need to sell need to sell 38 tickets38 tickets..
HomeworkHomework
Pages 553-555Pages 553-555
# 8, 12a-b, 14a-b, 18, 26, 28, 30, # 8, 12a-b, 14a-b, 18, 26, 28, 30, 3333
9-B
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