The month has flown by and in two days Marc and Kara will head home. They have had a terrific time in Boston and while they are happy to be heading home they are also sad to leave. For a final trip, Grandma and Grandpa decide to make a trip to the amusement park. Everyone is excited.
At the amusement park, different rides take a different number of tickets. Kara and Marc figure out that there are 9 rides that they would like to ride. These are their favorites and any others are just a bonus. Most often the teens will need three tickets for each ride. Each ride is a function of the number of tickets needed. Marc and Kara go up to the ticket booth and come back with 21 tickets each. If they each have 21 tickets, how many rides can they go on for those tickets?
Because each ride is a function of the number of tickets, you will need to know about functions to figure out this problem. Pay attention to this Concept and at the end you will be able to help with the day at the amusement park.
Guidance
Previously we worked with a given function rule or an equation. Now it is time to use those detective skills and figure out the rule from a given table.
Thinking like a detective looking for clues will help you write function rules.
One strategy that might be helpful in this task is to use guess and check to figure out the relationship between the values in the domain and the values in the range. Remember the \begin{align*}x\end{align*} values are the domain and the \begin{align*}y\end{align*} values are the range.
Identify the rule for this input-output table. Describe the rule in words.
Input number \begin{align*}(x)\end{align*} | Output number \begin{align*}(y)\end{align*} |
---|---|
10 | 1 |
20 | 2 |
30 | 3 |
40 | 4 |
Use guess and check to determine how to each pair of values is related.
First, notice that each output number (\begin{align*}y-\end{align*}value) is less than its corresponding input number (\begin{align*}x-\end{align*}value). There are two operations that help numbers become smaller. So, the rule will involve either subtraction or division.
Look for a subtraction rule first.
Consider the ordered pair (10, 1).
\begin{align*}10-1=9\end{align*}, so the rule could be to subtract 9 from each input number to find the corresponding output number. Check to see if that rule works for the other pairs of values in the table.
Consider the ordered pair (20, 2).
\begin{align*}20-1=19\end{align*}, not 2, so that rule does not work all the pairs of values in the table.
Look for a division rule next.
Consider the ordered pair (10, 1) again.
\begin{align*}10 \div 10=1\end{align*}, so the rule could be to divide each input number by 10 to find the corresponding output number. Check to see if that rule works for the other pairs of values in the table.
Consider the ordered pair (20, 2).
\begin{align*}20 \div 10=2\end{align*}, so the rule works for that ordered pair.
Consider the ordered pair (30, 3).
\begin{align*}30 \div 10=3\end{align*}, so the rule works for that ordered pair.
Consider the ordered pair (40, 4).
\begin{align*}40 \div 10=4\end{align*}, so the rule works for that ordered pair.
The rule for this function table is to divide each input number (\begin{align*}x-\end{align*}value) by 10 to find its corresponding output number (\begin{align*}y-\end{align*}value).
In this example, we wrote the function rule in words. We can also use an equation to express a function rule. Think about this past example.
The rule was “Divide each \begin{align*}x\end{align*} value by 10 and that will give you the \begin{align*}y\end{align*} value.”
We could write \begin{align*}\frac{x}{10}=y\end{align*}.
This would have been the same rule expressed as an equation.
Let’s look at another situation where we will write an equation to represent a function rule.
Identify the rule for this function table. Describe the rule in words. Then write an equation to represent the relationship between the pairs of values in the table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 5 |
2 | 10 |
3 | 15 |
4 | 20 |
Use guess and check to determine how each pair of values is related.
Notice that each \begin{align*}y-\end{align*}value is greater than its corresponding \begin{align*}x-\end{align*}value. So, the rule will involve either addition or multiplication.
Look for an addition rule first.
Consider the ordered pair (1, 5).
\begin{align*}1+4=5\end{align*}, so the rule could be to add 4 to each \begin{align*}x-\end{align*}value to find the corresponding y-value. Check to see if that rule works for the other pairs of values in the table.
Consider the ordered pair (2, 10).
\begin{align*}2+5=7\end{align*}, not 10, so that rule does not work all the pairs of values in the table.
Look for a multiplication rule next. Each \begin{align*}y-\end{align*}value in the table is 5 more than the previous \begin{align*}y-\end{align*}value, so the rule may involve multiplying by 5.
Consider the ordered pair (1, 5).
\begin{align*}1 \times 5=5\end{align*}, so the rule could be to multiply each \begin{align*}x-\end{align*}value by 5 to find the corresponding \begin{align*}y-\end{align*}value. Check to see if that rule works for the other pairs of values in the table.
Consider the ordered pair (2, 10).
\begin{align*}2 \times 5=10\end{align*}, so the rule works for that ordered pair.
Consider the ordered pair (3, 15).
\begin{align*}3 \times 5=15\end{align*}, so the rule works for that ordered pair.
Consider the ordered pair (4, 20).
\begin{align*}4 \times 5=20\end{align*}, so the rule works for that ordered pair.
The rule for this function table is multiply each \begin{align*}x-\end{align*}value by 5 to find its corresponding \begin{align*}y-\end{align*}value.
Now that we have a rule in words, let's write an equation to show the same relationship. Remember, to find each \begin{align*}y-\end{align*}value, you must multiply each \begin{align*}x-\end{align*}value by 5. So, the equation would be:
\begin{align*}y=x \cdot 5\end{align*} or \begin{align*}y=5x\end{align*}.
Yes it is. You will also become better at figuring out the rules over time.
One thing to consider is that sometimes, a function rule will involve more than one step. In other words, it will have two operations in it, not just one.
Look at each function table and write each rule as an equation.
Example A
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
5 | 10 |
7 | 14 |
9 | 18 |
Solution: \begin{align*}2x\end{align*}
Example B
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
3 | 5 |
4 | 7 |
6 | 11 |
Solution: \begin{align*}2x - 1\end{align*}
Example C
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
2 | 1 |
4 | 2 |
6 | 3 |
Solution: \begin{align*}\frac{x}{2}\end{align*}
Here is the original problem once again.
The month has flown by and in two days Marc and Kara will head home. They have had a terrific time in Boston and while they are happy to be heading home they are also sad to leave. For a final trip, Grandma and Grandpa decide to make a trip to the amusement park. Everyone is excited.
At the amusement park, different rides take a different number of tickets. Kara and Marc figure out that there are 9 rides that they would like to ride. These are their favorites and any others are just a bonus. Most often the teens will need three tickets for each ride. Each ride is a function of the number of tickets needed. Marc and Kara go up to the ticket booth and come back with 21 tickets each. If they each have 21 tickets, how many rides can they go on for those tickets?
The first thing to note is that each ride is a function of the number of tickets needed. We can call the ride \begin{align*}x\end{align*} and the ticket number \begin{align*}y\end{align*}.
You know from this lesson that one of the best ways to work with a function is through a function table. Let’s build one now.
\begin{align*}x\end{align*} Rides | \begin{align*}y\end{align*} Tickets |
---|
We can say that 1 ride is equal to 3 tickets. Let’s choose some other ride numbers and work our way up to 21 tickets.
\begin{align*}x\end{align*} Rides | \begin{align*}y\end{align*} Tickets |
---|---|
1 | 3 |
2 | 6 |
3 | 9 |
21 |
Look at the pattern. We can write an equation to show this pattern.
\begin{align*}y=3x\end{align*}
We want to figure out how many rides the teens can go on for 21 tickets. 21 is the \begin{align*}y\end{align*} in the equation. We can use what we have learned about solving equations to solve for \begin{align*}x\end{align*}. This will give us the number of rides they can go on for 21 tickets.
\begin{align*}21 &= 3x\\ 7 &= x\end{align*}
\begin{align*}x\end{align*} Rides | \begin{align*}y\end{align*} Tickets |
---|---|
1 | 3 |
2 | 6 |
3 | 9 |
7 | 21 |
The teens can go on 7 rides each for the 21 tickets. To ride all nine rides, they will need six more tickets each.
Vocabulary
- Function
- a set of ordered pairs in which one element corresponds to exactly one other element. Functions can be expressed as a set of ordered pairs or in a table.
- Domain
- the \begin{align*}x\end{align*} value of an ordered pair or the \begin{align*}x\end{align*} values in a set of ordered pairs.
- Range
- the \begin{align*}y\end{align*} value of an ordered pair or the \begin{align*}y\end{align*} values in a set of ordered pairs.
- Input-Output Table
- a way of showing a function using a table where the \begin{align*}x\end{align*} value is shown to cause the \begin{align*}y\end{align*} value through a function rule.
- Function Table
- another name for an input-output table.
- Function Rule
- a written equation that shows how the domain and range of a function are related through operations.
Guided Practice
Here is one for you to try on your own.
Identify the rule for this function table. Then write the rule.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 6 |
2 | 11 |
3 | 16 |
4 | 21 |
Answer
Use guess and check to determine how to each pair of values is related.
Notice that each \begin{align*}y-\end{align*}value is greater than its corresponding \begin{align*}x-\end{align*}value. So, the rule will involve either addition or multiplication.
Since each \begin{align*}y-\end{align*}value in the table is 5 more than the previous \begin{align*}y-\end{align*}value, the rule may involve multiplying by 5.
Look for a two-step rule that involves multiplying by 5.
Consider the ordered pair (1, 6).
\begin{align*}1 \times 5=5\end{align*} and \begin{align*}5+1=6\end{align*}, so the rule could be to multiply each input number by 5 and then to add 1 to find each \begin{align*}y-\end{align*}value. Check to see if that rule works for the other pairs of values in the table.
Consider the ordered pair (2, 11).
\begin{align*}2 \times 5=10\end{align*} and \begin{align*}10+1=11\end{align*}, so the rule works for that ordered pair.
Consider the ordered pair (3, 16).
\begin{align*}3 \times 5=15\end{align*} and \begin{align*}15+1=16\end{align*}, so the rule works for that ordered pair.
Consider the ordered pair (4, 21).
\begin{align*}4 \times 5=20\end{align*} and \begin{align*}20+1=21\end{align*}, so the rule works for that ordered pair.
The rule for this function table is to multiply each \begin{align*}x-\end{align*}value by 5 and then add 1 to find its corresponding \begin{align*}y-\end{align*}value.
Now that we have a rule in words, let's write an equation to show the same relationship. Remember, to find each \begin{align*}y-\end{align*}value, we must multiply each \begin{align*}x-\end{align*}value by 5 and then add 1. So, the equation would be
\begin{align*}y=5x+1\end{align*}
Video Review
This James Sousa video is about writing a function rule from an input - output table.
Practice
Directions: Determine the rule for each table. Use that rule to answer the questions.
Look at this table
Input number \begin{align*}(x)\end{align*} | Output number \begin{align*}(y)\end{align*} |
---|---|
1 | 3 |
2 | 4 |
3 | 5 |
4 | 6 |
1. Identify the rule.
2. Write an equation for the rule.
Look at this table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 2 |
2 | 4 |
3 | 6 |
4 | 8 |
3. Describe the rule in words.
4. Write an equation to represent the rule.
Look at this table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 3 |
2 | 5 |
3 | 7 |
4 | 9 |
5. Describe the rule in words.
6. Write an equation to represent the rule.
Directions: Solve each problem.
7. The total number of cupcakes Shakir can bake, \begin{align*}y\end{align*}, is a function of the number of batches of cupcake batter he makes, \begin{align*}x\end{align*}. This table shows the total number of cupcakes Shakir will bake if he makes 1, 2 or 3 batches of cupcake batter.
Number of Batches of Batter \begin{align*}(x)\end{align*} | Total Number of Cupcakes Baked \begin{align*}(y)\end{align*} |
---|---|
1 | 12 |
2 | 24 |
3 | 36 |
4 | ? |
5 | ? |
8. Write a rule in words to describe the relationship between the pairs of values in the table above.
9. Write an equation to describe the relationship between the pairs of values in the table above.
10. What is the total number of cupcakes that Shakir will bake if he makes 5 batches of batter?
For a concert in the auditorium, Ms. Walsh set up 10 chairs on the stage for the performers and 20 chairs in each row for the audience. This equation shows the relationship between \begin{align*}r\end{align*}, the number of rows of chairs Ms. Walsh set up and \begin{align*}t\end{align*}, the total number of chairs set up in the auditorium:
\begin{align*}t=10+20r\end{align*}.
11. Create a table to show the total number of chairs that would be set up in the auditorium if Ms. Walsh set up 0, 1, 2, 3, or 4 rows of chairs.
12. Identify the domain for the ordered pairs in the table.
13. Explain why the domain shown in the table is a reasonable domain for this function and why including a number less than 0 would not be reasonable.
Look at this table.
Input | Output |
---|---|
1 | 4 |
2 | 5 |
3 | 6 |
4 | 7 |
14. What is the rule?
15. Write an equation to represent the rule.