12.5: Graphing Functions
Introduction
Roller Coaster Speed
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 lesson is all about graphing functions. Pay close attention and at the end of this lesson you will be able to help Jana organize and graph her function.
What You Will Learn
By the end of this lesson you will be able to demonstrate the following skills:
- Graph linear functions in the coordinate plane.
- Distinguish between linear and non-linear functions.
- Use function graphs to relate perimeter, area and volume to linear dimensions of objects.
- Model and solve real-world problems involving patterns of change, using multiple representations of functions.
Teaching Time
I. Graph Linear Functions in the Coordinate Plane
In the last lesson you learned about functions. We actually didn’t really call them “functions,” but we called them input/output tables. Let’s look at what it means for the data in an input/output table to be a function.
What is a function?
A function is a set of data that has a specific relationship. One variable in the data set is related to or depends on a different variable in the same data set. Each input matches with only one output.
Let’s look at a table to look at this.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
0 | 2 |
1 | 4 |
2 | 6 |
3 | 8 |
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*} instead of input and output. They mean the same thing, but in mathematics as you work with functions, 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 input value and the \begin{align*}y\end{align*} is the output value. 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. This means that this table forms a function.
Here is another table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 5 |
1 | 7 |
3 | 9 |
4 | 13 |
Do you see something different here?
The \begin{align*}x\end{align*} value of 1 is connected two different "y" values at the same time. This is NOT a function. Because at least one input has multiple outputs (1 corresponds to both 5 and 7), this is not a function.
What does it mean when real life data forms a function?
It means that one variable depends on or is a function of the other variable in the data.
Let’s look at an example.
Example
Felix has a job cutting grass in the summer time. He earns $10.00 per lawn that he cuts.
This is an example of a function.
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 is a function of the number of lawns.
We can look at some data about Felix and then show how this forms a function.
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.
Here is our table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | $10 |
2 | $20 |
3 | $30 |
4 | $40 |
We can say that the amount of money that Felix earns is a function of the number of lawns that he mows.
We can also graph functions on the coordinate grid. We do this by using the values in each column to form our ordered pairs.
Notice that we have an \begin{align*}x\end{align*} value and a \begin{align*}y\end{align*} value. In an ordered pair we have an \begin{align*}x\end{align*} value and a \begin{align*}y\end{align*} value.
Let’s write this data as ordered pairs.
(1, 10)
(2, 20)
(3, 30)
(4, 40)
Now we can graph our data.
We create a graph by plotting the \begin{align*}x\end{align*} values (the number of dollars earned) on the \begin{align*}x\end{align*} axis and the \begin{align*}y\end{align*} values (the number of lawns) 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 function. Anytime that a graph forms a line like this it is called a linear graph-and a linear graph is a graph of a linear function.
In the next section you will learn more about linear and non-linear functions and their graphs.
Practice graphing the following functions.
1.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 5 |
2 | 10 |
3 | 15 |
4 | 20 |
2.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 8 |
2 | 6 |
3 | 4 |
4 | 2 |
3.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
2 | 4 |
4 | 6 |
6 | 8 |
10 | 12 |
Take a few minutes to check your work with a partner. Do your graphs show linear graphs?
II. Distinguish Between Linear and Non-Linear Functions
In the last section 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.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 10 |
2 | 8 |
3 | 6 |
4 | 4 |
Take a few minutes to check your work with a peer.
III. Use Function Graphs to Relate Perimeter, Area and Volume to Linear Dimensions of Objects
In an earlier lesson, you learned about how to calculate the perimeter, area and volume of different figures. Well now that you have learned about functions, we can apply functions to our work with perimeter, area and volume.
Perimeter, area and volume are a function of their dimensions.
What does this mean?
It means that when you know the dimensions of the figures, you can calculate the perimeter area and volume. We can also create a function table to show possible dimensions for certain areas, perimeters and volumes of figures.
Let’s start by looking at area.
Let’s say that I have the area of a rectangle. You know that we can calculate the area of a rectangle by multiplying the length times the width.
\begin{align*}A = lw\end{align*}
The length and width are the two variables. These can change, and depending on how they change, our area can also change.
What are the possible dimensions for a rectangle with an area of 36 square units?
How can we tackle this problem? Well, we can start by creating a table. We know that the length and width are going to be the variables, just like \begin{align*}x\end{align*} and \begin{align*}y\end{align*} were in the last section. Let’s create a table given this information.
\begin{align*}w\end{align*} | \begin{align*}l\end{align*} |
---|
Now we have to choose values for the table. We need to choose values that we can multiply together to give us an area of 36 sq. units. Everything goes back to the area of the figure given the dimensions.
We can use the formula for our rule. Watch how this works. If I put two in for the width, then the length has to be 18, because two times 18 is 36.
\begin{align*}w\end{align*} | \begin{align*}l\end{align*} |
---|---|
2 | 18 |
Next, we need to find three other values for dimensions that will give us 36 square units for an area.
\begin{align*}w\end{align*} | \begin{align*}l\end{align*} |
---|---|
2 | 18 |
3 | 12 |
4 | 9 |
6 | 6 |
What would a graph of this function look like?
We can graph this function and use \begin{align*}w\end{align*} and \begin{align*}l\end{align*} as our \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
Here is a graph of the function showing that a fixed area of 36 has several possible lengths and widths. Notice that it is a linear function.
What about perimeter?
We can create a function to match the dimensions of a rectangle with regards to perimeter too. Let’s say that we have a fixed perimeter of 12 units. We use the formula for perimeter of a rectangle to create our table of values.
\begin{align*}P = 2L + 2W\end{align*}
Next, we select a set of values for the width and then figure out the length using the formula. Remember that no matter which values we choose, that we have to end up with a perimeter of 12 units.
\begin{align*}w\end{align*} | \begin{align*}l\end{align*} |
---|---|
2 | 4 |
3 | 3 |
4 | 2 |
5 | 1 |
Now let’s graph the function and see if this is a linear function.
This is also a linear graph. A fixed perimeter of a rectangle forms a linear function.
What about volume?
Volume is the amount of space inside a three-dimensional figure.
We find the volume of a rectangular prism by multiplying the length, the width and the height.
\begin{align*}V = lwh\end{align*}
If we have a rectangular prism, we can keep the width and the height the same and only change the length. Watch what happens to the volume of the rectangular prism if we keep doubling the length of the prism.
Width = 4
Height = 2
\begin{align*}L\end{align*} | \begin{align*}V\end{align*} |
---|---|
4 | 32 |
8 | 64 |
16 | 128 |
How did we get these numbers?
We got them by using the formula for volume and by keeping the width and the height the same and we kept changing the length. In fact, we kept doubling the length.
What happened to the volume each time the length was doubled?
Each time the length was doubled, the volume also doubled.
Volume is a function of the relationship between the length, width and height of a prism.
Is this a linear function?
Notice that the increment that each value changes by is consistent and sequential. If we were to graph this function it would create a linear function.
Practice applying this information.
- Create a table for a rectangle with a fixed area of twenty-four square units. Then graph the results.
Check your table and graph with a friend. Is this a linear function?
Take a few notes on linear and non-linear functions and how to tell the difference before moving on to the next section.
Real Life Example Completed
Roller Coaster Speed
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 that are found in this lesson.
- 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.
- Perimeter
- the distance around the outside edge of a figure.
- Area
- the measure of the surface of a two-dimensional figure
- Volume
- the measure of the space contained inside a three-dimensional figure.
Technology Integration
James Sousa, Plotting Points on the Coordinate Plane
James Sousa, Graphing Equations by Plotting Points, Part 1
James Sousa, Graphing Equations by Plotting Points, Part 2
Resources
You can learn more about roller coasters at these websites.
www.wikipedia.org
www.cedarpoint.com
www.rcdb.com
Time to 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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