What Makes a
Function Linear
Lesson 1.3
1
What Is A Line?

Check out this list of definitions or
explanations to the question
 Define

Line
Today we look at what makes a function
linear.
2
Rate of Change

Consider example y = 3x + 5
 We
note the rate of change is constant
 This means it is a linear function
 It graphs as a straight line
y=3x + 5
3
Plotting Points on the Calculator

Use Data Matrix on Calculator

Choose APPS,
then 6 Data/Matrix
then NEW
4
Starting Up
Choose DATA
 Give it a variable
name for saving
in memory

5
Entering Data
Enter numeric values in the cells
 Enter a formula at the top

 Using
Cursor must be here
column name
Enter formula
here
6
Viewing Data
Note the results of the formula
 We can do further calculations
 We can also plot these points

7
Plotting Data
Choose F2 for Plot Setup
Screen
 Then F1 for Define

Choose line type
Specify the columns for
the X and Y values
8
Plotting Data
Goto the Y= Screen to turn off any
functions there
 Then specify ZoomData

 This
fits the window to the limits of the data
9
Plotting Data
Note the graph includes the points we had
in the data matrix
 It is a line-graph, the points are
represented by boxes

10
Another Example

Consider the following table of values
 Note the value of y
x
for any two pairs of
values
y
3
x
x  2
x  3
C1
x
C2
y
3
6
4
9
5
12
6
15
y  6
y  9
11
Example From Text
See Example 2, pg. 19
 Formula used for depreciation

Value of equipment = original value – $4000 * number of years

To generalize:
Dependent Qty = startValue + rateOfChange * independentQuantity
y  m x b
12
Family of Linear Functions

Slope = Rate of Change
y  m x b
y change in y
m  slope 

x change in x
 constant rate of change
y=3x + 5
View TI Nspire
filewhich demonstrates
the slope-intercept
formula
• Slope = m = 3
• y-intercept = b = 5
13
Family of Linear Functions

Calculating slope with two ordered pairs
(X2, Y2)
•
(X1, Y1)
•
x
y
Given two ordered pairs, (7,5) and
(-3,12). What is the slope of the line
through these two points?
y1  y2
 slope
x1  x2
5  12
7
   0.7
7  (3)
10
14
Warning
Not all functions which appear linear will
actually be linear!!
t
P
 Consider the set of
0 67.38
ordered pairs

 Graph
them
 Decide whether graph
is linear
 Check slope for different
pairs
1
2
3
4
5
6
69.13
70.93
72.77
74.67
76.61
78.60
15
Results
80
78
Graph appears
straight
 But …
rate of change is
not a constant

76
74
72
70
68
66
0
2
t
P
0
67.38
1
69.13
1.75
2
70.93
1.8
3
72.77
1.84
4
74.67
1.9
5
76.61
1.94
6
78.6
1.99
4
6
8
slope
16
Assignment
Lesson 1.3
 Page 24
 Exercises
1 – 5, 7, 9, 13, 15, 19, 21, 23

17