7.20: Linear and Non-Linear Patterns of Change
Have you ever thought of roller coasters? Marc loves them.
Marc loves roller coasters. He can’t wait to ride some of the roller coasters at an amusement park in New Hampshire. Marc thinks that the speed of the roller coaster is a function of its height.
After doing some research, here is what Marc discovers.
The Timber Terror Roller Coaster
Height
Speed
Kingda Ka Roller Coaster
Height
Speed
Top Thrill Dragster Roller Coaster
Height
Speed
Create a table and figure out if Marc's information is linear or non - linear.
This Concept will teach you how to solve real - world problems involving patterns of change.
Guidance
Linear functions can also allow us to represent real-world situations. Specifically, linear functions can help us understand situations in which two pairs of values are related by a rule.
Remember, we can represent the problem with a function table, an equation, or a graph. Representing a problem in more than one way can sometimes help us understand how to solve it.
This table shows how the total cost of buying tomatoes at the farmer's market changes depending on the number of pounds of tomatoes purchased.
Number of Pounds Purchased |
Total Cost in Dollars |
---|---|
1 | 2 |
2 | 4 |
3 | 6 |
4 | ? |
5 | ? |
a. Write an equation to describe the relationship between the pairs of values in the table above.
b. Create a graph to represent the relationship between the number of pounds of tomatoes purchased,
c. Determine the cost of buying 5 pounds of tomatoes at the farmer's market.
Consider part
Use guess and check to determine how to each pair of values is related.
For example, notice that each
Since each
Look for a rule that involves multiplying by 2.
Consider the ordered pair (1, 2).
Consider the ordered pair (2, 4).
Consider the ordered pair (3, 6).
So, the rule for this function table is: multiply each
Now that we have a rule in words, let's write an equation to show the same relationship. Remember, to find each
Next, consider part
First, let's consider how the graph for this function should look. We should use the horizontal axis to show the number of pounds,
We should use the vertical axis to show the total cost,
Consider how to number the axes. The
Next, we can plot the ordered pairs (1, 2), (2, 4) and (3, 6) and draw a line through them to create our graph. We can plot these values because they are the known values from the table.
The graph above represents the relationship between the number of pounds of tomatoes purchased,
Finally, consider part
One strategy for determining the total cost,
When
Now it's your turn to try a few. Find
Example A
When
Solution:
Example B
When
Solution:
Example C
When
Solution:
Here is the original problem once again.
Marc loves roller coasters. He can’t wait to ride some of the roller coasters at an amusement park in New Hampshire. Marc thinks that the speed of the roller coaster is a function of its height.
After doing some research, here is what Marc discovers.
The Timber Terror Roller Coaster
Height
Speed
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*}
Create a table and figure out if Marc's information is linear or non - linear.
First, let's create a table and then we can graph the data.
To create a table of Marc’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, Marc 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
- A pattern where one element of from the domain is paired with exactly one element from the range.
- Function Rule
- the pattern rule for a function.
- Linear Function
- a function that forms a straight line when graphed
- Non-Linear Function
- a function that does not form a straight line when graphed
Guided Practice
Here is one for you to try on your own.
True or false. For a graph to represent a linear change, each interval in the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} value must increase or decrease at the same interval.
Answer
True. This graph will be straight line. Otherwise, it is a non - linear function.
Video Review
Here is a video for review.
- This is a Khan Academy video on patterns and equations.
Practice
Directions: Look at each table and determine whether each pattern of change 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 |
This table shows how the total cost of buying gasoline at Gary's Gas Station changes depending on the number of gallons purchased.
Number of Gallons Purchased \begin{align*}(x)\end{align*} | Total Cost in Dollars \begin{align*}(y)\end{align*} |
---|---|
0 | 0 |
1 | 3 |
2 | 6 |
3 | ? |
11. Write an equation to describe the relationship between the pairs of values in the table above.
12. Create a graph to represent the relationship between the number of gallons purchased, \begin{align*}x\end{align*}, and the total cost, \begin{align*}y\end{align*}. Use the blank axes below to create your graph.
13. Determine the cost of buying 3 gallons of gasoline at Gary's Gas Station.
Franklin has a $10 bus card. Each time he rides the bus, $2 is deducted from his card. This equation shows the relationship between \begin{align*}x\end{align*}, the number of times he uses his card to ride the bus and \begin{align*}y\end{align*}, the number of dollars that are left on his card:
\begin{align*}y=10-2x\end{align*}.
14. Create a table to show how many dollars will be left on Franklin's bus card after he has used it for a total of 0, 1, 2, or 3 bus rides.
15. Create a graph to represent the relationship between the total number of bus rides Franklin uses the card to take, \begin{align*}x\end{align*}, and the number of dollars left on the card, \begin{align*}y\end{align*}. Use the blank axes below to create your graph.
16. If Franklin takes a total of 4 bus rides, how many dollars will be left on his bus card?
Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .Function Rule
A function rule describes how to convert an input value () into an output value () for a given function. An example of a function rule is .Linear Function
A linear function is a relation between two variables that produces a straight line when graphed.Non-Linear Function
A non-linear function is a function that does not form a line when graphed.Image Attributions
Description
Learning Objectives
Here you'll learn to model and solve real-world problems involving patterns of change and linear functions.