12.13: Linear and Non-Linear Function Distinction
Do you like roller coasters? Take a look at this dilemma.
Jana loves roller coasters. She can’t wait to ride some of the roller coasters at the amusement park for the class trip. Jana is so curious about roller coasters that she starts to do some research about them. For example, Jana wonders whether or not the speed of the roller coaster is connected to the height of the roller coaster or the length of the roller coaster. She thinks that the speed of the roller coaster is a function of its height.
After doing some research, here is what Jana discovers.
The Timber Terror Roller Coaster
Height \begin{align*}= 85 \ ft\end{align*}
Speed \begin{align*}= 55 \ mph\end{align*}
Kingda Ka Roller Coaster
Height \begin{align*}= 456 \ feet\end{align*}
Speed \begin{align*}= 128 \ mph\end{align*}
Top Thrill Dragster Roller Coaster
Height \begin{align*}= 420 \ ft.\end{align*}
Speed \begin{align*}= 120 \ mph\end{align*}
Jana wants to show how this data appears in a chart. She wants to be able to prove that the speed of the roller coaster is a function of its height.
This Concept is all about graphing functions. Pay close attention and at the end of this Concept you will be able to help Jana organize and graph her function.
Guidance
In the last Concept, you learned to identify a linear function. Let’s identify a linear function now.
What is a linear function?
A linear function has a graph that is straight line.
Let’s look at this table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
0 | 2 |
1 | 4 |
2 | 6 |
3 | 8 |
Notice that each \begin{align*}x\end{align*} value has a \begin{align*}y\end{align*} value that gets larger as it goes up. As one value increases the dependent or \begin{align*}y\end{align*} value increases too. It does this in a sequential way. We can tell that this graph will form a straight line.
Let’s be sure that it does. Here is the graph of this function.
That’s a great question.
What is a non-linear function?
A non-linear function is a function where the data does not increase or decrease in a systematic or sequential way. In short, a non-linear function does not form a straight line when it is graphed.
Let’s look at a non-linear function in a table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 3 |
2 | 5 |
3 | 4 |
4 | 9 |
Do you notice anything different about this function?
The data does not move in a sequential way. This graph will not form a straight line.
Let’s graph this function to be sure. Here is the graph of a non-linear function.
We could connect these points, but it does not change the fact that this is a non-linear function.
Practice identifying whether each represents a linear or a non-linear function.
Example A
Solution: Non - Linear Function
Example B
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 10 |
2 | 8 |
3 | 6 |
4 | 4 |
Solution: Linear Function
Example C
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 8 |
2 | 6 |
3 | 4 |
4 | 2 |
Solution: Linear Function
Now back to the roller coaster.
Here is the original problem once again. Reread the problem and then work on creating a table and function graph of Jana’s data.
Jana loves roller coasters. She can’t wait to ride some of the roller coasters at the amusement park for the class trip. Jana is so curious about roller coasters that she starts to do some research about them. For example, Jana wonders whether or not the speed of the roller coaster is connected to the height of the roller coaster or the length of the roller coaster. She thinks that the speed of the roller coaster is a function of its height.
After doing some research, here is what Jana discovers.
The Timber Terror Roller Coaster
Height \begin{align*}= 85 \ ft\end{align*}
Speed \begin{align*}= 55 \ mph\end{align*}
Kingda Ka Roller Coaster
Height \begin{align*}= 456 \ feet\end{align*}
Speed \begin{align*}= 128 \ mph\end{align*}
Top Thrill Dragster Roller Coaster
Height \begin{align*}= 420 \ ft.\end{align*}
Speed \begin{align*}= 120 \ mph\end{align*}
To create a table of Jana’s data we must use the height as one variable and the speed as the other. Here is a table of our data.
\begin{align*}H\end{align*} | \begin{align*}S\end{align*} |
---|---|
85 | 55 |
420 | 120 |
456 | 128 |
You can see that as the height increases so does the speed. Using this information, Jana can conclude that the speed of a roller coaster is a function of its height.
Let’s create a graph of the function.
Notice that this graph is a non-linear graph. Even though the speed increases with the height of the roller coaster, the interval that it increases is not even. Therefore, the graph of this function is non-linear.
Vocabulary
Here are the vocabulary words in this Concept.
- Function
- one variable is dependent on another. One variable matches exactly one other value.
- Linear Function
- the graph of a linear function forms a straight line.
- Non-Linear Function
- the graph of a non-linear function does not form a straight line.
Guided Practice
Here is one for you to try on your own.
Non - linear or linear?
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
2 | 4 |
4 | 6 |
6 | 8 |
10 | 12 |
Answer
This pattern does not follow a linear rule. It will not form a straight line when graphed. It is a non - linear function.
Video Review
Here are videos for review.
Khan Academy: Recognizing Linear Functions
Khan Academy: Exploring Non-Linear Relationships
Practice
Directions: Look at each table and determine whether the function is linear or non-linear.
1.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
0 | 2 |
1 | 3 |
2 | 5 |
4 | 4 |
2.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 3 |
2 | 5 |
3 | 7 |
4 | 9 |
3.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
2 | 6 |
3 | 9 |
5 | 15 |
6 | 18 |
4.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
2 | 3 |
3 | 4 |
6 | 7 |
8 | 9 |
5.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
8 | 4 |
6 | 12 |
2 | 8 |
0 | 0 |
6.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
0 | 3 |
1 | 4 |
2 | 5 |
6 | 9 |
7.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
5 | 11 |
4 | 9 |
3 | 7 |
2 | 5 |
8.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 7 |
3 | 4 |
2 | 9 |
5 | 8 |
9.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 3 |
2 | 6 |
4 | 12 |
6 | 18 |
10.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
4 | 2 |
5 | 3 |
6 | 5 |
7 | 1 |
Directions: Now use each table in 1 – 10 and graph each function. You should have 10 graphs for this section. Number these graphs 11 – 20. If the graph is a linear graph, then please connect the points with a line.
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Image Attributions
Here you'll learn to distinguish between linear and non - linear functions.