4.2 Linear Relations4.2 Linear Relations

Math 9

Linear Relations

• In this unit, we are going to look at relationships between variables.

• These relationships, when graphed, produce straight lines. – We call these relationships LINEAR RELATIONS.

• Every line is made up of an infinite number of coordinates (x, y). When the coordinates are close enough together they look like a line!

• Before we can graph a line, we must be able to determine the coordinates that make up that line. – We can do this by using a table of values.

1. Identify 3 coordinates that are on the line.2. Using these coordinates create a table of values that

represents this graph. Include 2 more points in your table.

1. Identify 3 coordinates that are on the line.2. Using these coordinates create a table of values that

represents this graph. Include 2 more points in your table.

x y

What do the arrows mean on this line?How many other coordinates are on this line?What do the arrows mean on this line?How many other coordinates are on this line?

( _____ , _____ )

( _____ , _____ )

( _____ , _____ )

( _____ , _____ )

( _____ , _____ )

( _____ , _____ )

More about lines…

• Every line (in the world!) can be described by using an equation (just like we use an equation for area, perimeter, etc.).

• The equations all relate 2 variables – an independent and a dependent variable (we’ll define these later).

x y

-1 -1

0 0

1 1

2 2

3 3

Can you determine an equation that relates x and y?Can you determine an equation that relates x and y?

What can you say about the relationship

between x and y.

What can you say about the relationship

between x and y.

The table of values below shows a relationship between 2 variables. Create graphs using the table of values. The table of values below shows a relationship between 2 variables. Create graphs using the table of values.

Can you determine an equation that relates x and y?Can you determine an equation that relates x and y?

What can you say about the relationship

between x and y.

What can you say about the relationship

between x and y.

Try this one… create a graph from this table of values… Try this one… create a graph from this table of values…

x y

0 -5

1 -2

2 1

3 4

4 7

2 2y x

Sometimes we are just given an equation and asked to graph it. Here’s an example:

Graph: Graph

:

x y

-1

0

1

2

How can we get coordinates by using this equation?

How can we get coordinates by using this equation?

Let’s look at it a bit further…

• How many points do you need in order to graph a line?

• Why is it a good idea to have more than the minimum number of points?

• From your graph, determine 2 more points on the line. By using the equation, PROVE they are on the line.

2 2y x

Let’s Define a Few Words!• Dependent Variable

– A variable whose value is determined by the value of the independent variable (eg. In this example, the value of the perimeter is dependent on the number of rectangle. So, the PERIMETER is the DEPENDENT variable.) This is graphed on the y-axis.

• Independent Variable– A variable whose value determines the value of

another variable (the dependent variable). (eg. In this example, the number of the rectangle determines the perimeter, so the NUMBER is the INDEPENDENT variable.) This is graphed on the x-axis. HINT: the left hand column of a table of values is always the independent variable and is graphed on the x-axis.

• Dependent Variable– A variable whose value is determined by the value of

the independent variable (eg. In this example, the value of the perimeter is dependent on the number of rectangle. So, the PERIMETER is the DEPENDENT variable.) This is graphed on the y-axis.

• Independent Variable– A variable whose value determines the value of

another variable (the dependent variable). (eg. In this example, the number of the rectangle determines the perimeter, so the NUMBER is the INDEPENDENT variable.) This is graphed on the x-axis. HINT: the left hand column of a table of values is always the independent variable and is graphed on the x-axis.

Let’s Define a Few Words!• Discrete Data

– Data that can be counted. DO NOT connect the points of a discrete data graph. This graph is an example of DISCRETE data.

• Continuous Data– Data that cannot be counted. CONNECT the points

of a continuous data graph in a straight line.• Linear Relation

– When two variables are related and the graph produces a straight (linear) line.

Example (if needed)• The first 4 rectangles in a pattern are shown below. The

pattern continues. Each small square has a side length of 1 cm.

• The perimeter of the rectangle is related to the rectangle number.

• Express the relationship between the perimeter and the rectangle number using words, a table of values, an equation and a graph.

#1 #3#2

Using a Table of Values

Rectangle Number, n

Perimeter, p (cm)

1

2

3

4

4 2(2) 8

4 2(3) 10

“As the rectangle number increases by 1, the perimeter increases by 2 cm.”

“As the rectangle number increases by 1, the perimeter increases by 2 cm.”

4 2(1) 6

4 2(4) 12

What is an equation that

represents this data?

What is an equation that

represents this data?

2 4P n 2 4P n

Using a Graph

Graph of P against n

4

6

8

10

0

2

4

6

8

10

12

0 1 2 3 4 5

n (number of squares)

P (

per

imet

er)

Clearly label each axis (in

this case P and n). Also, label your scale (go up by 1’s, 2’s,

5’s, etc.)

Clearly label each axis (in

this case P and n). Also, label your scale (go up by 1’s, 2’s,

5’s, etc.)

Don’t forget your title!Don’t forget your title!

To determine whether or not to connect the points, ask yourself if the points in between

‘matter’.

To determine whether or not to connect the points, ask yourself if the points in between

‘matter’. Independent variable is on the x-axisIndependent variable is on the x-axis

Dependent variable is on the y-axisDependent variable is on the y-axis

“Real Life” Example #1• The student council is planning to hold a dance. The

profit in dollars is 4 times the number of students that attend, minus $200 for the cost of the music. The maximum number of students that can attend is 200.

1. Write an equation that relates the profit to the number of students who attend.

2. Create a table of values for this relation.

3. Graph the data in the table. Does it make sense to join the points?

4. How many students have to attend to make a profit?

Equation:Equation:

Example #1 Cont’d…

Is the relation LINEAR? Justify your answer.

Is the relation LINEAR? Justify your answer.

Do you connect the points? Why

or why not?

Do you connect the points? Why

or why not?

Number of Students

Profit

0

25

50

100

200

How many students have to attend to

make a profit?

How many students have to attend to

make a profit?

Estimate how many students attended the

dance if they made $80?

Estimate how many students attended the

dance if they made $80?

Example #2

• The table of values shows the cost of renting DVD’s at an online store.

1. Graph the data..

2. Use the table to describe the pattern (using an equation) in the rental costs. How is this pattern shown in the graph?

Number of DVD’s

Rented (d)Cost (C)

1 $3.50

2 $7.00

3 $10.50

4 $14.00

5 $17.50

Example #2 Cont’d…

How do you decide what to put

on the x- and y-axis?

How do you decide what to put

on the x- and y-axis?

Do you connect the points? Why

or why not?

Do you connect the points? Why

or why not?Is the relation LINEAR? Justify your answer.

Is the relation LINEAR? Justify your answer.

Number of DVD’s

Rented (d)Cost (C)

1 $3.50

2 $7.00

3 $10.50

4 $14.00

5 $17.50

What is the equation that

represents this relationship?

What is the equation that

represents this relationship?

AssignmentAssignment

• Page 170/1714 -9,11,14, 15

• COMPLETE ON GRAPH PAPER!

Using Words• What happens to the perimeter as you go from rectangle 1,

to rectangle 2 and so on?• Be specific when you describe the relationship!

• What happens to the perimeter as you go from rectangle 1, to rectangle 2 and so on?

• Be specific when you describe the relationship!