Representing Proportional Relationships

8.EE.6, 8.F.1

Essential Question?How can you use tables, graphs, mapping diagrams, and equations to represent proportional situations?

Common Core Standard:

8.F.1 ─ Define, evaluate, and compare functions.Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

8.EE.6 ─ Understand the connections between proportional relationships, lines, and linear equations.Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at b.

Objectives:

• Understand that a function is a rule that assigns to each input exactly one output.

• Identify whether a relationship is a function from a diagram, table of values, graph, or equation.

Curriculum Vocabulary

Function (función):A relationship between an independent variable, x, and a dependent variable, y, where each value of x (input) has one and only one value of y (output).

Relation (relación):Any set of ordered pairs.

Input (entrada):A number or value that is entered.

Output (salida):The number or value that comes out from a process.

The diagram below shows the function “add 2.”

Input = 3 Function: Add 2 Output = 5

There is only one possible output for each input.

The function “add 2” is expressed in words.

It can also be:

• written as the equation y=x+2• represented by a table of values• represented as a mapping diagram• shown as a graph.

FUNCTIONS

Look at the following table:

For EACH INPUT THERE IS EXACTLY ONE OUTPUT.

You can notice that there isNO REPETITION in the INPUT column.

This table represents a function.

IDENTIFYING FUNCTIONS

INPUT OUTPUT

10 68

13 73

14 75

18 68

21 74

Look at the following table:

For EACH INPUT THERE IS MORE THAN ONE OUTPUT.

You can notice that there isREPETITION in the INPUT column.

This table DOES NOT represent a function.

IDENTIFYING FUNCTIONS

INPUT OUTPUT

3 9

3 10

5 25

5 26

7 49

What are some relationships that are functions?

Each coin of American currency is assigned one specific value in dollars. For example, the value of a penny is always $0.01. In this function, an ordered pair relates the name of a coin and its value in dollars.

Coin Penny Nickel Dime Quarter Half-Dollar

Dollar Value 0.01 0.05 0.10 0.25 0.50

FUNCTIONS

There is a ONE to ONE relationship!

Most mathematical functions include ordered pairs of numbers. For example, a 120-pound person burns about 65 calories per mile while walking. The table below shows how many calories the person would burn walking different numbers of miles.

Miles (input) 1 2 3 4 5 6

Calories (output) 65 130 195 260 325 390

The input is the number of miles walked.The rule (function) is to multiply the number of miles by 65.The output is the number of calories burned.

FUNCTIONS

FUNCTIONS

A basketball coach gives the starting players a game jersey. At the same time he measures the players’ heights. This relationship is a function. For each jersey number, he records only one player’s height.

Let’s examine the following function:

Player’s Jersey Number (input) 10 13 14 18 21

Players height in inches (output) 68 73 75 68 74

There is a ONE to ONE relationship!This represents a FUNCTION!

FUNCTIONS

Let’s REVERSE the input and output from the previous table:

What are some relationships that are not functions?

There is NOT a ONE to ONE relationship!

THIS IS NOT A FUNCTION!

Players height in inches (input) 68 73 75 68 74

Player’s Jersey Number (output) 10 13 14 18 21

Notice there is REPETION IN THE INPUT.The input 68 has more than one output.

REPRESENTING FUNCTIONS

There are 4 (FOUR) ways to represent a function that we will explore:

1. TABLE

2. MAPPING DIAGRAM

3. EQUATION

4. GRAPH

We have already seen how we can represent a relationship using a table. Now let’s create a MAPPING DIAGRAM.

68

73

74

75

10

13

14

18

21

Input: Jersey Number Output: Height

REPRESENTING FUNCTIONS

Player’s Jersey Number (input) 10 13 14 18 21

Players height in inches (output) 68 73 75 68 74

When we reversed the input and output we already discovered that the new relationship is not a function.

68

73

74

75

10

13

14

18

21

Input: Height Output: Jersey Number

REPRESENTING FUNCTIONS

Players height in inches (input) 68 73 75 68 74

Player’s Jersey Number (output) 10 13 14 18 21

REPRESENTING FUNCTIONS

The third way we can represent a function is by writing an EQUATION.

Let’s examine the following table:

Do you notice a pattern? Do you think you can come up with a rule?

If you can see a pattern that is ALWAYS THE SAME, that means there is a CONSTANT RATE OF CHANGE.

Numberof

Hours Worked

Money Earned

1 12

2 24

3 36

4 48

5 60

PROPORTIONAL RELATIONSHIPS

A PROPORTIONAL RELATIONSHIP is a relationship between two quantities in which the ration of one quantity to the other is CONSTANT.

In this example the change is always$12 earned for every 1 hour worked.

We can set up a fraction:

or In this example it would be:

or

Numberof

Hours Worked

Money Earned

1 12

2 24

3 36

4 48

5 60

1 12

1 12

1 12

1 12

PROPORTIONAL RELATIONSHIPS

A PROPORTIONAL RELATIONSHIP can be described by an equation in the form , where is a number called the

CONSTANT OF PROPORTIONALITY.

If , then

In our example the constant of proportionality, k, is or 12 .

The equation would be:

Numberof

Hours Worked

Money Earned

1 12

2 24

3 36

4 48

5 60

1 12

1 12

1 12

1 12

PROPORTIONAL RELATIONSHIPS

It is important to note that all PROPORTIONAL RELATIONSHIPS MUST contain the ordered pair (0,0) [THE ORIGIN] as part of the table. If (0,0) does not appear, the function is NOT proportional.Be careful to EXTEND THE TABLE, to be sure.Example: Does the table represent a proportional relationship?

Is there a constant rate of change?

Do you see (0,0) on the table?

Input 2 4 6 8 10

Output 5 10 15 20 25

PROPORTIONAL RELATIONSHIPSLet’s see if there is a constant rate of change.

Input 2 4 6 8 10

Output 5 10 15 20 25

We see that for each time the output decreases by 5,the input decreases by 2.

There IS a constant rate of change!

-2

-5-5

-2-2

-5-5

-2

PROPORTIONAL RELATIONSHIPSLet’s extend the table to see if (0,0) is part of the table.

Input 2 4 6 8 10

Output 5 10 15 20 25

Input 2 4 6 8 10

Output 5 10 15 20 25

0

0

Now we see that (0,0) is part of the table.

Since there is a constant rate of change and (0,0) is part of the table, this IS PROPORTIONAL.

We can write the equation as or

-2

-5-5

-2-2

-5-5

-2

-5

-2

PROPORTIONAL RELATIONSHIPS

Does this table represent a proportional relationship?

Input 1 2 3 4 5

Output 1 4 9 16 25

Is there a constant rate of change?

NOThis is NOT a proportional relationship.

PROPORTIONAL RELATIONSHIPSDoes this table represent a proportional relationship?

Input 1 2 3 4 5

Output 15 20 25 30 35

Is there a constant rate of change?YES

If you extend the table, will (0,0) be part of it?NOThis is NOT a proportional relationship.

PROPORTIONAL RELATIONSHIPSSo far we have seen a proportional relationship represented as:

• A TABLE• A MAPPING DIAGRAM• An EQUATION

Input Output

1 3

2 6

3 9

4 12

5 15

1

2

3

4

5

3

6

9

12

15

𝑦=3 𝑥

PROPORTIONAL RELATIONSHIPSThe fourth way to represent a proportional relationsip is as

• A GRAPHThe best way to graph a relationship is to use the table.Create ordered pairs and plot them.

Input Output

1 3

2 6

3 9

4 12

5 15

(1,3)

(2,6)

(3,9)

(4,12)

(5,15)

REPRESENTING FUNCTIONSSince we know there is a CONSTANT RATE OF CHANGE, we can connect the dots with a STRAIGHT LINE.

PROPORTIONAL RELATIONSHIPS

Input Output

1 3

2 6

3 9

4 12

5 15

Let’s examine the graph of a proportional relationship and anon-proportional relationship and see if we can draw a conclusion.

PROPORTIONAL RELATIONSHIP NON-PROPORTIONAL RELATIONSHIP

Input Output

1 2

2 3

3 4

4 5

5 6

What is the main difference between the two graphs?

The graph of a PROPORTIONAL RELATIONSHIP

is a LINE that PASSES THROUGH THE ORIGIN!

PROPORTIONAL RELATIONSHIPSWe have now seen a PROPORTIONAL RELATIONSHIP represented as:

• A TABLE• CONSTANT rate of change• CONTAINS the ordered pair (0,0)

• A MAPPING DIAGRAM• Shows a ONE to ONE relationship

• An EQUATION• Takes the form where is the constant of proportionality

• A GRAPH• A LINE that passes through the ORIGIN

Input Output

1 3

2 6

3 9

4 12

5 15

1

2

3

4

5

3

6

9

12

15

𝑦=3 𝑥