2.4 Using Linear Models
September 18, 2008
2­4 Using Linear Models
Objective:
To write linear equations that model real world data
To make predictions from linear models
Jul 22­2:02 PM
1
2.4 Using Linear Models
September 18, 2008
Check Skills You'll Need
Find the change in x and the change in y between each pair of points.
1) (­0.2, 9) and (3.4, 7.3)
2) (10, 17) and (11.5, 13.5)
Evaluate each function for the given values.
4
1
3) f(x) = x ­ 2 for x = ­3, 0, 2
3
Jul 22­2:02 PM
2
2.4 Using Linear Models
September 18, 2008
Modeling Real World Data
You can write linear equations to model real­world problems.
Example: Transportation
Jacksonville, Florida has an elevation of 12 ft. above sea level. A hot­air balloon taking off from Jacksonville rises 50 ft/min. Write an equation to model the balloon's elevation as a function of time. Graph the equation. Interpret the intercept at which the graph intersects the vertical axis. Jul 22­2:02 PM
3
2.4 Using Linear Models
September 18, 2008
Jacksonville, Florida has an elevation of 12 ft. above sea level. A hot­air balloon taking off from Jacksonville rises 50 ft/min. Write an equation to model the balloon's elevation as a function of time. Graph the equation. Interpret the intercept at which the graph intersects the vertical axis. Relate:
balloon's elevation = (rate)(time) + starting elevation
Define:
Let h = the balloon's elevation
Let t = time (in minutes) since the hot­air balloon lifted off
Write:
h = (50)(t) + 12
h = 50t + 12
h
The h­intercept is (0, 12).
The t­coordinate, 0, represents the time at the start of the trip.
30
The h­coordinate, 12, represents the elevation of the balloon at the start of the trip.
.
20
10 (0, 12)
2
4
6
t
Jul 22­2:02 PM
4
2.4 Using Linear Models
September 18, 2008
A spring has a length of 8 cm when a 20­g mass hangs at the bottom end. Each additional gram stretches the spring another 0.15 cm. Write an equation to model the length y of the spring as a function of the mass x of the attached weight.
Step 1:
Identify the data points (20, 8) and (21, 8.15) as (x1, y1) and (x2, y2).
Step 2:
Find the slope of the line.
Step 3:
Use one of the points and the point­slope form to
write an equation for the line.
y = 0.15x + 5
Jul 22­2:02 PM
5
2.4 Using Linear Models
September 18, 2008
Using the previous equation, what mass could be needed to stretch the spring to a length of 9.5 cm?
y = 0.15x + 5
9.5 = 0.15x + 5
4.5 = 0.15x
30 = x
Jul 22­2:02 PM
6
2.4 Using Linear Models
September 18, 2008
A scatter plot is a graph that relates two different sets of data by plotting the data as ordered pairs. Strong,
negative
No
A trend line is a line that approximates the relationship between the data sets of a scatter plot. You can use a trend line to make predictions.
Jul 22­2:02 PM
7
2.4 Using Linear Models
September 18, 2008
A woman is considering buying a 1999 car. She researches prices for various years of the same model car and records the data in a table.
Model Year
Prices
2000
2001
$5784 $6810
2002
$8237
2003
$9660
2004
$10,948
$5435
$7751
$9127
$10,455
j
$6207
Let x represent model year (in years since 1999)
Let y represent the price of the car
Draw a scatter plot
Aug 10­10:16 AM
8
2.4 Using Linear Models
Model Year
Prices
September 18, 2008
2000
2001
$5784 $6810
2002
$8237
2003
$9660
2004
$10,948
$5435
$7751
$9127
$10,455
j
$6207
Price (in thousands)
8
6
4
2
2
4
6
8
years (since 1999)
Aug 10­10:16 AM
9
2.4 Using Linear Models
Model Year
Prices
September 18, 2008
2000
2001
$5784 $6810
2002
$8237
2003
$9660
2004
$10,948
$5435
$7751
$9127
$10,455
j
$6207
Price (in thousands)
8
6
Draw a trend line
4
2
2
4
6
8
years (since 1999)
Aug 10­10:16 AM
10
2.4 Using Linear Models
Model Year
Prices
September 18, 2008
2000
2001
$5784 $6810
2002
$8237
2003
$9660
2004
$10,948
$5435
$7751
$9127
$10,455
j
$6207
Price (in thousands)
8
Write the equation of the line
6
4
2
2
4
6
8
years (since 1999)
Aug 10­10:16 AM
11
2.4 Using Linear Models
September 18, 2008
If the car's asking price is $4,200, is this price reasonable?
Aug 10­10:16 AM
12
2.4 Using Linear Models
September 18, 2008
Homework
Homework
Page 81
#'s: 1 ­ 11 odd, 12, 13
2.1 ‐ 2.3 Quiz MONDAY!!!!
Aug 10­10:16 AM
13