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

x 3 4 6 9 y 4 6 10 18

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

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.

x 0 2 5 9 y 0 64 160 288

Press , select Plot1, turn the plot on, choose scatterplot, set Xlist and Ylist

Press , choose “EDIT”, put the x-values in L1 and the y-values in L2

Press 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

(b) x 0.3 1.3 2.2 3.3 4.1 y 5.6 2 0.92 0.77 0.51