Learning goals:
I will be able to use first differences to determine if a relation is linear.
I will be able to apply the characteristics of linear relations to solve problems.
At the Amusement Park,
The students have been having a fantastic time at the amusement park! Here is some information about tickets.
If the booklet had 6 tickets, then you can go on two rides. If the booklet had 12 tickets in it, then you could go on 4 rides. The number of rides is a function of the number of tickets. Using this chart can help the managers and the amusement park design new booklets of tickets.
\begin{align*}x\end{align*} Tickets | \begin{align*}y\end{align*} Rides |
---|---|
3 | 1 |
6 | 2 |
9 | 3 |
12 | 4 |
We can create a visual display of this data. How do we do this?
This is where graphing functions is important. A graph can show the relationship between the \begin{align*}x\end{align*} value and the \begin{align*}y\end{align*} value. In this Concept, you will learn about graphs. We will come back to this problem at the end of the Concept.
Guidance
Example A
Let’s look at a table of values to examine this problem. Let's calculate the first differences.
To calculate first differences,
1. check that the x values go up at a constant rate, ie. 1,2,3,4 or 2,4,6,8 etc.
2. subtract each pair of y values and put in third column of the table - see below.
3. if the first differences are all the same, then the relation is linear.
4. if the first differences are not all the same, then the relation is non-linear.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
First Differences |
---|---|---|
0 | 2 | -- |
1 | 4 | 4 - 2 = 2 |
2 | 6 | 6 - 4 = 2 |
3 | 8 | 8 - 6 = 2 |
Do you see something different in this table?
In this table we use the letters \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. They mean the same thing as the other variables, but in mathematics as you will use \begin{align*}x\end{align*} and \begin{align*}y\end{align*} more often. We can get used to seeing them in our work here.
Here the \begin{align*}x\end{align*} is the independent variable and the \begin{align*}y\end{align*} is the dependent variable. The \begin{align*}y\end{align*} value depends on the \begin{align*}x\end{align*} value. They go together. You can see that each value of the \begin{align*}x\end{align*} column matches with only ONE value of the \begin{align*}y\end{align*} column.
When we calculated the first differences, we determined that they were all equal to "2". When the first differences are all the same, we know that the relation is linear.
Here is another table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} | First Differences |
---|---|---|
1 | 5 | -- |
2 | 7 | 7 - 5 = 2 |
3 | 9 | 9 - 7 = 2 |
4 | 13 | 13 - 9 = 4 |
Do you see something different here?
The first differences are NOT all the same. This relation is non-linear.
Example B
Felix has a job cutting grass in the summer time. He earns $10.00 per lawn that he cuts.
The amount of money that Felix makes is related to the number of lawns that he cuts. If Felix cuts 10 lawns, then he will make $100.00. The amount of money depends on the number of lawns.
We can look at some data about Felix.
Felix cut the following lawns on four different days.
Day 1 = 1 lawn = $10.00
Day 2 = 2 lawns = $20.00
Day 3 = 3 lawns = $30.00
Day 4 = 4 lawns = $40.00
How can we organize this data in a table?
Well, the number of lawns would be the \begin{align*}x\end{align*} value and the amount of money earned would be the \begin{align*}y\end{align*} value. The \begin{align*}x\end{align*} is the value that can be counted on or depended on and the \begin{align*}y\end{align*} value changes depending on the \begin{align*}x\end{align*} value. We can calculate the first differences to determine if the relation is linear or non-linear.
Here is our table.
\begin{align*}x\end{align*} number of lawns |
\begin{align*}y\end{align*} money earned |
First Differences |
---|---|---|
1 | $10 | -- |
2 | $20 | 20 - 10 = 10 |
3 | $30 | 30 - 20 = 10 |
4 | $40 | 40 - 30 = 10 |
We can say that the amount of money that Felix earns depends on the number of lawns that he mows.
Now we can graph our data.
We create a graph by plotting the \begin{align*}x\end{align*} values (the number of lawns) on the \begin{align*}x\end{align*} axis and the \begin{align*}y\end{align*} values (the amount of money earned) on the \begin{align*}y\end{align*} axis.
Wow! This graph forms a line!
Yes it does. This graph forms what we call a linear relation.
Now back to the dilemma about tickets at the amusement park. Here is the original problem once again.
The students have been having a fantastic time at the amusement park! Here is some information about tickets.
If the booklet had 6 tickets, then you can go on two rides. If the booklet had 12 tickets in it, then you could go on 4 rides. The number of rides is dependent on the number of tickets. Using this chart can help the managers and the amusement park design new booklets of tickets.
When calculating the first differences, remember to check that your x-values go up by the same amount and then subtract y-values.
\begin{align*}x\end{align*} Tickets | \begin{align*}y\end{align*} Rides | First differences |
---|---|---|
3 | 1 | -- |
6 | 2 | 2 - 1 = 1 |
9 | 3 | 3 - 2 = 1 |
12 | 4 | 4 - 3 = 1 |
We can create a visual display of this data. How do we do this?
Here is a graph representing the data from the table.
Notice that this is a linear graph showing the relationship between rides and tickets.
Vocabulary
Here are the vocabulary words in this Concept.
- First Differences
subtracting subsequent y-values to see if they change by a constant amount; if constant, then the relation is linear; if not constant, the relation is non-linear.
- Linear
- the graph of a linear relation forms a straight line.
Non-Linearthe graph of a non-linear relation does NOT form a straight line
Guided Practice
Here is one for you to try on your own.
Let’s look at a table of values and see how we can represent the function on a coordinate grid.
On a coordinate plane, graph the linear function that is represented by the ordered pairs in the table below. Remember when calculating first differences, first check to see if the x-values go up by the same amount each time.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} | First Differences |
---|---|---|
-4 | 5 | -- |
-2 | 3 | 3 - 5 = -2 |
0 | 1 | ? |
2 | -1 | ? |
4 | -3 | ? |
Answer
Did you calculate all the first differences? They are all equal to -2. We can conclude that since the first differences are constant (all the same) that this is a linear relation.
To check, we can plot those five points on the coordinate plane. Then connect them as shown below.
Notice that the graph is a straight line. That is because this is a linear relation.
Video Review
Here is a video for review.