Section 9B Linear Modeling

1
Section 9BLinear Modeling

• Pages 542-553

2
Linear Functions
9-B

• A Linear Function changes by the same absolute
  amount for each unit of change in the input
  (independent variable).
• A Linear Function has a constant rate of change.

ExamplesStraightown population as a function of
time. Postage cost as a function of
weight. Pineapple demand as a function of price.
3
First Class Mail a linear function
9-B
Weight Postage cost
1 oz 0.37
2 oz 0.60
3 oz 0.83
4 oz 1.06
5 oz 1.29
6 oz 1.52
7 oz 1.75
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First Class Mail a linear function
9-B
Weight Postage cost Difference
1 oz 0.37
2 oz 0.60
3 oz 0.83
4 oz 1.06
5 oz 1.29
6 oz 1.52
7 oz 1.75
5
First Class Mail a linear function
9-B
Weight Postage cost Difference
1 oz 0.37
2 oz 0.60 0.23
3 oz 0.83
4 oz 1.06
5 oz 1.29
6 oz 1.52
7 oz 1.75
6
First Class Mail a linear function
9-B
Weight Postage cost Difference
1 oz 0.37
2 oz 0.60 0.23
3 oz 0.83 0.23
4 oz 1.06
5 oz 1.29
6 oz 1.52
7 oz 1.75
7
First Class Mail a linear function
9-B
Weight Postage cost Difference
1 oz 0.37
2 oz 0.60 0.23
3 oz 0.83 0.23
4 oz 1.06 0.23
5 oz 1.29
6 oz 1.52
7 oz 1.75
8
First Class Mail a linear function
9-B
Weight Postage cost Difference
1 oz 0.37
2 oz 0.60 0.23
3 oz 0.83 0.23
4 oz 1.06 0.23
5 oz 1.29 0.23
6 oz 1.52
7 oz 1.75
9
First Class Mail a linear function
9-B
Weight Postage cost Difference
1 oz 0.37
2 oz 0.60 0.23
3 oz 0.83 0.23
4 oz 1.06 0.23
5 oz 1.29 0.23
6 oz 1.52 0.23
7 oz 1.75
10
First Class Mail a linear function
9-B
Weight Postage cost Difference
1 oz 0.37
2 oz 0.60 0.23
3 oz 0.83 0.23
4 oz 1.06 0.23
5 oz 1.29 0.23
6 oz 1.52 0.23
7 oz 1.75 0.23
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First Class Postage
9-B

12
First class postage a linear function
9-B

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First class postage a linear function
9-B

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First class postage a linear function
9-B

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We define rate of change of a linear function
by
where (x1,y1) and (x2,y2) are any two ordered
pairs of the function.
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Slope rate of change
9-B

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Linear Functions
9-B

• A linear function has a constant rate of change
• and a straight line graph.
• The rate of change slope of the graph.
• The greater the rate of change, the steeper the
  slope.
• positive slope negative slope

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Example Price-Demand Function
9-B

• A linear function is used to describe how the
  demand for pineapples varies with the price.
• (2, 80 pineapples) and (5, 50 pineapples).
• Find the rate of change (slope) for this function
  and then graph the function.
• independent variable price
• dependent variable demand for pineapples

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Example Price-Demand Function
9-B
(2, 80 pineapples) and (5, 50 pineapples)
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Example Price-Demand Function
9-B

• (2, 80 pineapples) and (5, 50 pineapples).
• To graph a linear function you need 2 things
• two points or
• slope and one point

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Example Price-Demand Function
9-B
(2, 80 pineapples) and (5, 50 pineapples).
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Example Price-Demand Function
9-B
(2, 80 pineapples) and (5, 50 pineapples).
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General Equation for a Linear Function
9-B

• dependent initial value
  (slope)independent
• y initial value (slope)x
• (Initial value occurs when the independent
  variable 0.)
• y mx b or
• y b mx
• m slope
• b y-intercept
• (The line goes through the
  point (0,b).)

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Example
9-B

• dep. variable initial value (slope) indep.
  variable
• 
  
• 
  
• slope -10 pineapples/
• initial value 100 pineapples
• Demand 100 - 10(price)
• D 100 10p

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Example
9-B

• 
  
• 
  
• Demand 100 - 10(price)
• D 100 10p
• Check 2 100 - 102 80 pineapples
• 5 100 - 105 50 pineapples

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old example The initial population of
Straightown is 10, 000 and increases by 500
people per year.
Graph
Data Table
t Pf(t)
0 f(0)10,000
5 f(5)12,500
10 f(10)15,000
15 f(15)17,500
20 f(20)20,000
40 f(40)30,000
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old example The initial population of
Straightown is 10, 000 and increases by 500
people per year.
t Pf(t)
0 10,000
5 12,500
10 15,000
15 17,500
20 20,000
40 30,000
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)
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Example First class postage
9-B
Slope .23/ounce initial
value 0.14
Weight Postage cost
1 oz 0.37
2 oz 0.60
3 oz 0.83
4 oz 1.06
5 oz 1.29
6 oz 1.52
7 oz 1.75
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Example First Class Postage
9-B

• Slope .23/ounce
• initial value 0.14
• Postage 0.14 0.23(weight)
• P 0.14 0.23w
• Check 1 ounce 0.14 0.231 0.37
• 6 ounces 0.14 0.236 1.52

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Example
9-B

• The world record time in the 100-meter butterfly
  was 53.0 seconds in 1988. Assume that the record
  falls at a constant rate of 0.05 seconds per
  year. What does the model predict for the record
  in 2010?
• dependent variable world record time (R)
• independent variable is time, t (years) after
  1988.
• Slope 0.05 seconds initial value 53.0
  seconds
• Record time 53.0 0.05(t years after 1988)
• R 53 0.05t
• Record time in 2010 53 - .05(22) 51.9 seconds

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Example
9-B
Suppose you were 20 inches long at birth and 4 ft
tall on your tenth birthday. Create a linear
equation that describes how your height varies
with age. independent variable age
(years) dependent variable height (inches) Two
points (0, 20) (10, 48) Initial value 20
inches Height 20 2.8t t years
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Example
9-B
Fines for Certain PrePayable Violations
Speeding other than residence zone, highway work
zone and school crosswalk 5.00 per MPH over
speed limit plus processing fee (51.00) and
local fees (5.00) independent variable miles
over speed limit dependent variable fine
() Initial value 56.00 Slope 5.00 Fine
56 5(your speed-speed limit)
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Example
9-B
Mrs. M. was given a ticket for doing 52 mph in a
zone where the speed limit was 35 mph. How much
was her fine? Fine 55 5(her
speed-35) Fine 56 5(52-35)
56 5(17) 141
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Example
9-B
Fines for Certain PrePayable Violations
Speeding in a residence zone 200 plus 7.00
per MPH over speed limit (25 mph), plus
processing fee (51.00) and local fees
(5.00) independent variable miles over speed
limit dependent variable fine () Initial value
256.00 Slope 7.00 Fine 256 7(your
speed-25)
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Example
9-B
The Psychology Club plans to pay a visitor 75 to
speak at a fundraiser. Tickets will be sold for
2 apiece. Find a linear equation that gives the
profit/loss for the event as it varies with the
number of tickets sold. independent variable
number of tickets sold dependent variable
profit/loss () (0, -75) slope 2 (
rate of change in ticket price) Profit -75
2(number of tickets) P -75 2n
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Example
9-B
How many people must attend for the club to break
even? P -75 2n 0 -75 2n 75 2n 37.5
n Cant sell half a ticket -- so well need to
sell 38 tickets.
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9-B

• Homework
• Pages 553-555
• 8, 12a-b, 14a-b, 18, 26, 28, 30, 33