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Linear Functions

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A function is a relationship between sets of numbers. The first set of numbers determines the value of the second set of numbers.

The value of the first set of numbers corresponds to one and only one value for the second set of numbers.

A linear function is a function that represents a straight line.

Do you remember the input/output tables like this one?

Input
Output
5
7
10
12
1
3
6
8

This table represents a rule, i.e. add 2.

This is a function because the relationship between the input and the output is that the output is always 2 more than the input.

We could write the given relationship (or function) like this:

y = x + 2

where, x represents the input and the y represents the output.

Now let's replace x and y with the values given in the table.

y = x + 2

7 = 5 + 2

12 = 10 + 2

3 = 1 + 2

8 = 6 + 2

We can also write a function in different ways.

  • As an input/output table
  • In words (such as "the output is always two more than the input")
  • As an equation (such as, y> = x + 2)
  • As a graph

To write an equation for the function, follow these steps:

  • Determine the rule.
  • Write the rule as an equation.
    x represents the input and y represents the output.

Let's go through some examples.

Write an equation for the function shown in this table:

Input
Output
2
20
5
50
7
70
3
30
 

Solution:

First of all, identify the rule.

We can see that each time the input is being multiplied by 10. So the rule is "multiply by 10".

Now write the function as an equation.

y = 10 ? x

where y represents the output and x represents the input.

Write an equation for the function shown in this table:

Input
Output
17
12
18
13
17
7
6
1
 

Solution:

First of all, identify the rule.

We can see that each time 5 is subtracted from the input. So the rule is "subtract 5".

Now write the function as an equation.

y = x - 5

where y represents the output and x represents the input.

The pattern in this table is that the input is multiplied by 2 then 3 is added to it. Write an equation for the function.

Input
Output
0
3
1
5
2
7
3
9
 

Solution:

We already know that the rule is to multiply by 2 then add 3.

Write the function as an equation.

y = 2x + 3

where y represents the output and x represents the input.

When Veronica goes to the arcade, it always costs $5 to get in plus whatever tokens she buys.

 

Solution:

First of all, let's figure out the rule from the statement.

Total cost of Veronica is cost of tokens plus $5.

Now to write this function or rule as an equation, let input (x) be the cost of her tokens and the output (y) be the total cost.

y = x + 5

Remember we do not use units in the rule.

Summary
  • A function is a relationship between sets of numbers. The first set of numbers determines the value of the second set of numbers.
  • The value of the first set of numbers corresponds to one and only one value for the second set of numbers.
  • A linear function is a function that represents a straight line.
  • A function can be written in different ways:
    • As an input/output table
    • In words (such as "the output is two more than the input")
    • As an equation (such as, y = x + 2)
    • As a graph

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