3.3 Modeling Data With Linear Functions
A table of data is linear if there is a constant average rate of change between all pairs of points.
Testing for Linearity
• to test if a table of data is linear, calculate the average rate of change between each
consecutive pair of points
•
if the rate of change is constant, the data represents a linear function
•
if not, then it is not a linear function
x
0
2
5
9
y
0
64
160
288
64 ! 0
= 32,
2!0
160 ! 64
288 !160
= 32, !!!
= 32
5! 2
9!5
So the table represents a linear function.
x
3
4
6
9
y
4
6
10
18
6!4
= 2,
4!3
10 ! 6
= 2,
6!4
18 !10 8
=
9!6 3
So this table is not a linear function.
Example 1
Determine whether or not each of the following tables represents a linear function.
(a)
x
2
4
6
8
y
12
17
21
25
(b)
x
0
3
5
6
9
y
4
13
19
22
31
Finding a Formula for a Table that is Linear
•
the constant average rate of change found when determining that the table is linear is the
slope of the line
•
use this slope and any one point from the table, write the equation using the point-slope
form, then solve for “y” to get the function equation
Continuing our earlier example:
x
0
2
5
9
y
0
64
160
288
64 ! 0
= 32,
2!0
160 ! 64
288 !160
= 32, !!!
= 32
5! 2
9!5
So the table represents a linear function.
Now using the point (2, 64) we get:
y ! 64 = 32 ( x ! 2 )
" y ! 64 = 32x ! 64
" y = 32x
Note, we could have used any of the 4 points given in the table and we would get the
same final answer.
Example 2
Show that the table represents a linear function and then give a formula for the function.
x
2
5
7
12
y
8.5
15.7
20.5
32.5
Example 3
The table below shows the cost C, in dollars, for a company to produce n widgets. Show that
table is represents a linear function, find the formula for this function, and then represent in
function notation the cost to produce 875 widgets.
n
200
250
300
350
C
7900
9650
11400
13150
Plotting a Table of Data
We will walk through plotting the data from the first example.
Press
Press
Press
x
0
2
5
9
y
0
64
160
288
, select Plot1, turn the plot on, choose scatterplot, set Xlist and Ylist
, choose “EDIT”, put the x-values in L1 and the y-values in L2
to see the scatterplot
Example 4
Plot each of the following tables of data. Does the plot look linear?
(a)
x
1.3
2.5
3.3
4.2
5.1
y
2.6
2.3
2
1.8
1.5
x
0.3
1.3
2.2
3.3
4.1
y
5.6
2
0.92
0.77
0.51
(b)