1

What Makes a Function Linear

Lesson 1.3

2

What Is A Line?

Check out this list of definitions or explanations to the questionDefine Line

Today we look at what makes a function linear.

3

Rate of Change

Consider example y = 3x + 5We note the rate of change is constantThis means it is a linear function It graphs as a straight line

y=3x + 5

4

Plotting Points on the Calculator

Use Data Matrix on Calculator

Choose APPS, then 6 Data/Matrixthen NEW

5

Starting Up

Choose DATA Give it a variable

name for savingin memory

6

Entering Data

Enter numeric values in the cells Enter a formula at the top

Using column nameCursor must be here

Enter formula here

7

Viewing Data

Note the results of the formula We can do further calculations We can also plot these points

8

Plotting Data

Choose F2 for Plot SetupScreen

Then F1 for Define

Choose line type

Specify the columns for the X and Y values

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Plotting Data

Goto the Y= Screen to turn off any functions there

Then specify ZoomDataThis fits the window to the limits of the data

10

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

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Another Example

Consider the following table of values Note the value of

for any two pairs of values

C1x

C2y

3 6

4 9

5 12

6 15

y

x

2x 6y 3x 9y 3

y

x

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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

$20,000

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Family of Linear Functions

Slope = Rate of Change

change in y

change in x

constant rate of change

y m x b

ym slope

x

y=3x + 5• Slope = m = 3

• y-intercept = b = 5

View TI Nspire filewhich demonstrates

the slope-intercept formula

View TI Nspire filewhich demonstrates

the slope-intercept formula

14

Family of Linear Functions

Calculating slope with two ordered pairs

•

•(X1, Y1)

(X2, Y2)

xy

1 2

1 2

y yslope

x x

Given two ordered pairs, (7,5) and (-3,12). What is the slope of the line

through these two points?

5 12 70.7

7 ( 3) 10

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Warning

Not all functions which appear linear will actually be linear!!

Consider the set of ordered pairsGraph themDecide whether graph

is linearCheck slope for different

pairs

t P

0 67.38

1 69.13

2 70.93

3 72.77

4 74.67

5 76.61

6 78.60

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Results

Graph appearsstraight

But …rate of change is not a constant

66

68

70

72

74

76

78

80

0 2 4 6 8

t P slope

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

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Assignment

Lesson 1.3 Page 24 Exercises

1 – 5, 7, 9, 13, 15, 19, 21, 23