12.4: Functions
Introduction
The Car Wash
Mrs. Hawk’s sixth grade class was so motivated by the idea of taking a coach bus on their class trip that they had 100% turn out for the car wash on Saturday. The students gathered their supplies and washed cars for most of the day. The car wash started at 9 am and continued until 2 pm.
The students figured out that they needed to earn $244.92. To make the math easier, they rounded up to $245.00. At $5.00 a car, they needed to wash 49 cars to make enough money for the bus.
The car wash was a busy place. At the beginning there weren’t any cars, but between 9 am and 10 am the class washed 5 cars. From 10 to 11, the class washed 10 cars, from 11 to 12 the class washed 15 cars and from 12 – 1 the class washed 20 cars.
Toby kept track of all of this information in his notebook. He created a chart to show how the number of cars washed changed throughout the day.
\begin{align*}&0 \qquad 0\\ &1 \qquad 5\\ &2 \qquad 10\\ &3 \qquad 15\\ &4 \qquad 20\end{align*}
Toby can see a pattern in the data, can you? In this lesson you will learn how to write rules for patterns. Pay close attention and at the end of this lesson you will have chance to write a rule that matches this table.
What You Will Learn
By the end of this lesson you will be able to demonstrate the following skills:
- Write an expression for an input-output table.
- Evaluate a given function rule for an input-output table.
- Write a function rule from an input-output table.
- Model and solve real-world problems involving patterns of change.
Teaching Time
I. Write an Expression for an Input-Output Table
Patterns are everywhere in life. They exist in nature and in machinery and even in temperatures. Detecting patterns is one of the things that mathematicians and scientists do every day. They look for patterns in the way that things are made or created or counted and then they can draw conclusions based on those patterns.
A pattern functions according to a rule. In this lesson, we are going to be looking at different patterns and at how to decipher and write rules for patterns.
What is a pattern?
A pattern is something that repeats in a specific way. A pattern functions according to a rule. The rule tells us how the pattern repeats.
We can look at patterns in nature-for example the number of leaves on a flower or the number of branches on a tree are special patterns.
Let’s look at an example of a pattern.
Example
2, 4, 6, 8, 10.....
Once you have a pattern, we can establish a rule about the pattern. This pattern counts by two’s. We could say that we add two to each previous term to get the next term in the pattern.
How can we write this so that anyone could understand the rule?
In this example, we could use a variable to represent the terms in the list. Let’s use \begin{align*}x\end{align*}.
\begin{align*}x=\end{align*} term in the pattern
By term we mean the numbers 2, 4, 6 and so on.
Next, we can add more to the variable. Since we add two to each term to get the next term, then we can say that \begin{align*}x\end{align*} plus two is the rule.
Rule: \begin{align*}x+2\end{align*}
Now let’s check the rule to be sure that it works for each term in the list.
2, 4, 6, 8, 10...
If I take 2 and substitute it for \begin{align*}x\end{align*} then 2 + 2 = 4, so the rule works.
If I take 4 and substitute it for \begin{align*}x\end{align*} then 4 + 2 = 6, so the rule works.
If I take 6 and substitute it for \begin{align*}x\end{align*} then 6 + 2 = 8, so the rule works.
Is there an easier way to figure this out?
Yes. We can use a table. We call it an input/output table.
Input | Output |
---|---|
2 | 4 |
4 | 6 |
6 | 8 |
8 | 10 |
Let’s see if our rule \begin{align*}x+2\end{align*} works for this table.
A term has been put into the table, that is the input. Then a term comes out, that is the output. The rule tells us what happened to the input to equal the output.
Does the rule \begin{align*}x+2\end{align*} work for each term in the table?
Yes it does. Two can be added to each term in the input column to equal the output column.
You can write rules by examining the patterns in input/output tables.
Let’s look at an example.
Example
Input | Output |
---|---|
0 | 0 |
1 | 3 |
2 | 6 |
3 | 9 |
What happened to the input to get the output?
This is where we can look at figuring out a rule. It is a little like deciphering a puzzle. You have to think of what happened to one term to equal another term.
The term in the input column was multiplied by 3 to get the number in the output column. This is the rule for this table.
We can write the rule as an expression.
If the input column is \begin{align*}x\end{align*}, then \begin{align*}3x\end{align*} is the rule for this table.
Rule \begin{align*}= 3x\end{align*}
Sometimes rules are a bit more complicated. Sometimes, there can be two operations in a rule.
Example
Input | Output |
---|---|
3 | 7 |
4 | 9 |
5 | 11 |
7 | 15 |
What is the rule of this table? What happened to the input to get the output?
This is tricky, but if you look for patterns you will see that the input was multiplied by two and then one was added. We can write the rule as an expression.
If you think of the input as a variable, we can write a rule for this table that looks like this.
Rule \begin{align*}= 2x+1\end{align*}
We call the input-output relationship of terms a function.
A function is when one variable or terms depends on another according to a rule. There is a special relationship between the two variables of the function where each value in the input applies to only one value in the output.
These rules that we have been writing we can call function rules, because they explain how the function operates. Here are some hints for writing function rules.
Hints for Writing Function Rules
- Decipher the pattern of the function. What happened to the input to get the output?
- Write the rule as an expression.
Think of the input as a variable.
Then write the operations used with this variable.
This will explain the function rule.
Look at these input-output tables and write each rule as an expression.
1.
Input | Output |
---|---|
10 | 6 |
9 | 5 |
8 | 4 |
7 | 3 |
2.
Input | Output |
---|---|
2 | 4 |
4 | 8 |
6 | 12 |
7 | 14 |
3.
Input | Output |
---|---|
0 | 5 |
1 | 6 |
2 | 7 |
4 | 9 |
Take a few minutes to check your work with a partner. Is each rule written as an expression?
II. Evaluate a Given Function Rule for an Input-Output Table
In the last section you had to figure out the function rules for each table. In this section, you will be given function rules and you must work to determine whether or not the rule is a rule for the table.
Example
Is \begin{align*}x + 4\end{align*} a rule for this function?
Input | Output |
---|---|
2 | 5 |
3 | 6 |
4 | 7 |
5 | 8 |
No. It is not. Look at the input. Each term in the input became the term in the output when 3 was added to it.
Our rule states that four was added. Therefore, this is not a viable rule.
Example
Is \begin{align*}5x\end{align*} a rule for this function?
Input | Output |
---|---|
20 | 100 |
10 | 50 |
5 | 25 |
1 | 5 |
Yes it is. In this case, each term in the input was multiplied by five to get the term in the output. Therefore this rule does work for this table.
Practice a few of these on your own. Figure out if each rule makes sense for the input-output table.
1. \begin{align*}4x\end{align*}
Input | Output |
---|---|
2 | 10 |
3 | 15 |
5 | 25 |
6 | 30 |
2. \begin{align*}2x-1\end{align*}
Input | Output |
---|---|
2 | 3 |
3 | 5 |
4 | 7 |
6 | 11 |
Take a few minutes to discuss your answers with a neighbor.
III. Write a Function Rule From an Input-Output Table
You have had some practice writing simple rules from input-output tables. Next, we are going to work on writing rules that are a little more challenging.
To work on these input-output tables, you will need to use all of your detective skills.
Write a rule that represents the input-output table.
Example
Input | Output |
---|---|
12 | 6 |
10 | 5 |
8 | 4 |
6 | 3 |
4 | 2 |
What rule could we write to represent what happened to the input to equal the output?
If you look, you will see that each term of the input was divided by two to get the output. We can use a variable for the input.
Rule: \begin{align*}\frac{a}{2}\end{align*}
This rule will work for each value in the table so it is a rule for this input-output table.
Example
Input | Output |
---|---|
3 | 5 |
5 | 9 |
7 | 13 |
8 | 15 |
10 | 19 |
What rule could we write to represent this function?
Here two operations were performed. The input value was multiplied by two and then one was subtracted. We can use a variable for the input and write the rule.
Rule: \begin{align*}2x-1\end{align*}
Real Life Example Completed
The Car Wash
Here is the original problem once again. Use what you have learned to write a rule for the pattern of cars washed. Be sure to reread the problem first and underline any important information.
Mrs. Hawk’s sixth grade class was so motivated by the idea of taking a coach bus on their class trip that they had 100% turn out for the car wash on Saturday. The students gathered their supplies and washed cars for most of the day. The car wash started at 9 am and continued until 2 pm.
The students figured out that they needed to earn $244.92. To make the math easier, they rounded up to $245.00. At $5.00 a car, they needed to wash 49 cars to make enough money for the bus.
The car wash was a busy place. At the beginning there weren’t any cars, but between 9 am and 10 am the class washed 5 cars. From 10 to 11, the class washed 10 cars, from 11 to 12 the class washed 15 cars and from 12 – 1 the class washed 20 cars.
Toby kept track of all of this information in his notebook. He created a chart to show how the number of cars washed changed throughout the day.
\begin{align*}&0 \qquad 0\\ &1 \qquad 5\\ &2 \qquad 10\\ &3 \qquad 15\\ &4 \qquad 20\end{align*}
Toby can see a pattern in the data, can you?
Each number in the left hand column shows the time that passed.
In the beginning there weren’t any cars.
Then in the first hour the students washed 5 cars.
In the second hour, they washed 10 cars.
In the third hour, they washed 15 cars.
In the fourth hour, they washed 20 cars.
If we wanted to write a rule for the pattern, what happened to the input to get the output?
The input was multiplied by 5.
The rule for the number of cars washed per hour is \begin{align*}5x\end{align*}.
Given this rule, how many cars can we predict will be washed in the fifth hour?
Write down your prediction and check it with a friend.
If each car paid $5.00, how much money did the students make in five hours?
75 \begin{align*}\times\end{align*} 5 \begin{align*}=\end{align*} $375.00
The students are very excited! They will be able to take the coach bus to the amusement park!!
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Pattern
- a series of pictures, numbers or other symbols that repeats in some way according to rule.
- Function
- one variable depends on the other and there is only one output for each input in a function.
- Input-Output Table
- A table that shows how a value changes according to a rule.
Technology Integration
James Sousa, Introduction to Functions, Part 1
James Sousa, Introduction to Functions, Part 2
James Sousa, Example of Writing a Cost Function and Completing a Table of Values
Other Videos:
- http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/5_Patterns/index.html – This is a fun video on examining and understanding patterns.
Time to Practice
Directions: Write an expression for each input-output table. Use a variable for the value in the input column of the table.
1.
Input | Output |
---|---|
1 | 4 |
2 | 5 |
3 | 6 |
4 | 7 |
2.
Input | Output |
---|---|
2 | 4 |
3 | 6 |
4 | 8 |
5 | 10 |
3.
Input | Output |
---|---|
1 | 3 |
2 | 6 |
4 | 12 |
5 | 15 |
4.
Input | Output |
---|---|
9 | 7 |
7 | 5 |
5 | 3 |
3 | 1 |
5.
Input | Output |
---|---|
8 | 12 |
9 | 13 |
11 | 15 |
20 | 24 |
6.
Input | Output |
---|---|
3 | 21 |
4 | 28 |
6 | 42 |
8 | 56 |
7.
Input | Output |
---|---|
2 | 5 |
3 | 7 |
4 | 9 |
5 | 11 |
8.
Input | Output |
---|---|
4 | 7 |
5 | 9 |
6 | 11 |
8 | 15 |
9.
Input | Output |
---|---|
5 | 14 |
6 | 17 |
7 | 20 |
8 | 23 |
10.
Input | Output |
---|---|
4 | 16 |
5 | 20 |
6 | 24 |
8 | 32 |
Directions: Go back through the tables and rules for number 1 – 10. Use each rule to calculate the output if the input is 12.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
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