2.4 Linear Functions and Models
I have dreams of orca whales and owls, but I wake up in fear.
-Regina Spektor
Linear Model Applications
A linear function is of the form
f (x)  mx  b
Suppose that a company just purchased a fleet of new cars for its sales force at a
cost of $28,000 per car. The company chooses to depreciate each vehicle using
the straight-line method over 7 years. This means that each car will depreciate by
$4000 per year. This can be represented by the following function:

V ( x)  4000 x  28000
Where x represents number of years.
c) When will the book value of each
a) Graph the linear function
(window 0 ≤ 𝑦 ≤ 30000, 0 ≤ 𝑥 ≤ 10) car be $8000? (show work)
b) Find the book value of each
car after 3 years._________
With the graph showing:
2nd  CALC (above TRACE)Value enter value for x  ENTER
Linear Model Applications
The quantity of a good is the amount of a product that a company is willing to
make available for sale at a given price. The quantity demanded of a good is the
amount of a product that consumers are willing to purchase at a given price.
Suppose that the quantity supplied, S, and the quantity demanded, D, of cell
phones each month are given by the following functions:
𝑺 𝒑 = 𝟔𝟎𝒑 − 𝟗𝟎𝟎
𝑫(𝒑) = −𝟏𝟓𝒑 + 𝟐𝟖𝟓𝟎
a) Graph the functions on the same graph (window 0 ≤ 𝑥 ≤ 100, 0 ≤ 𝑦 ≤ 3500).
Sketch the graph.
Linear Model Applications
b) Find the equilibrium price – the price at which the quantity supplied equals the
quantity demanded. ___________
With the graph showing:
2nd  CALC (above TRACE)  Intersect  Enter, Enter, Enter
What is the amount supplied at the equilibrium price? ___________
Label the equilibrium price and quantity on your graph.
c) Determine the prices for which quantity supplied is greater than quantity
demanded. (Show work)
Linear Model Applications
In baseball, the on-base percentage for a team represents the percentage of time
that the players safely reach base. The data given in the table represent the
number of runs scored, y, and the on-base percentages, x, for teams in the
National League during the 2003 baseball season.
Team
On-Base
Percent.
Runs
Scored
Team
On-Base Runs
Percent. Scored
Atlanta
34.9
907
Chic. Cubs
32.3
724
St. Louis
35.0
876
Arizona
33.0
717
Colorado
34.4
853
Milwaukee
32.9
714
Houston
33.6
805
Montreal
32.6
711
Phili.
34.3
791
Cincinnati
31.8
694
San Fran.
33.8
755
San Diego
33.3
678
Pittsburgh 33.8
753
NY Mets
31.4
642
Florida
751
Los Angeles 30.3
574
33.3
Linear Model Applications
a) Use your graphing calculator to create a scatter plot of the data.
Enter the data:
• STAT  Edit (make sure the lists are cleared before you start)
• Type in your data into each list. Use arrow keys to move between lists
Create the scatterplot:
• Hit Y=  use arrows to highlight ‘Plot1’  ENTER
• Hit ZOOM  9
b) Find the equation for the line of best fit: ________________________
Choose a linear regression model:
• STAT  arrow over to highlight CALC  option 4 (LinReg)  ENTER
• Use the data the calculator gave you to put the equation into slope-intercept
form (y = mx+b)
Graph the line of best fit
• VARS  option 5 (Statistics)  arrow over to highlight EQ  option 1 (RegEQ)
• Hit ZOOM  9
Linear Model Applications
c) If a team had an on-base percentage of 34.0, predict the number of runs they
scored during the season. _______________
With the graph showing:
• 2nd  CALC (above TRACE)  option 1 (value)
• Enter in the value for x, hit ENTER
d) If a team scores 760 runs in a season, what might have been their on-base
percentage? (show work)
2.4 Linear Functions and Models
Homework #20:
p. 103 #31-39 odd, 51-55 odd
I have dreams of orca whales and owls, but I wake up in fear.
-Regina Spektor