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Linear and Non-Linear Patterns of Change

Understand patterns of change as they appear on a graph.

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Linear and Nonlinear Patterns of Change
License: CC BY-NC 3.0

Some ocean birds dive from high above to catch fish swimming near the surface of the ocean. Samuel’s favorite ocean bird is called a tern. After much observation, Samuel figures out that the equation \begin{align*} h=19-16.1t^2\end{align*}, where \begin{align*}t\end{align*} is in seconds and \begin{align*}h\end{align*} is in feet, approximately models the height of a tern on its descent into the ocean to catch a fish.

Is this function linear or nonlinear? Can you plot points using this equation and sketch a graph of the tern’s descent into the ocean?

In this concept, you will learn to model and solve real-world problems involving patterns of change and linear functions.

Solving Problems Involving Patterns of Change

Functions can be used to model real-world data. Sometimes this data is linear, and can be modeled by a linear function, and sometimes the data is nonlinear, and thus cannot be modeled by a linear function.

Let’s look at an example.

The table below 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 Ponds Purchased \begin{align*}(x)\end{align*} Total Cost in Dollars \begin{align*}(y)\end{align*}
1 2
2 4
3 6
4 ?
5 ?
  1. Assuming the pattern presented continues write an equation which models the relationship between the number of pounds purchased and the total cost.
  2. Create a graph to represent the relationship between the number of pounds of tomatoes purchased, \begin{align*}x\end{align*}, and the total cost, \begin{align*}y\end{align*}.
  3. Determine the cost of buying 5 pounds of tomatoes at the farmer’s market.

Consider part a first.

You can see that as \begin{align*}x\end{align*} increases by 1, \begin{align*}y\end{align*} increases by 2. You may also notice that \begin{align*}y\end{align*} is twice as much as \begin{align*}x\end{align*}.

Since you assume that this pattern continues, the rule for this function is: multiply each \begin{align*}x\end{align*}-value by 2 to find its corresponding \begin{align*}y\end{align*}-value.

Now, translate those words to an equation. The equation is \begin{align*}y = 2x\end{align*}.

Next, consider part b.

You need to make a graph that represents the function \begin{align*}y = 2x\end{align*}.

First, plot the coordinates \begin{align*}(1, 2), (2, 4)\end{align*} and \begin{align*}(3, 6) \end{align*} from the table. Then, draw a line through the points.

License: CC BY-NC 3.0

The graph above represents the relationship between the number of pounds of tomatoes purchased, \begin{align*}x\end{align*}, and the total cost, \begin{align*}y\end{align*}.

Finally, consider part c.

There are two ways to approach finding what \begin{align*}y\end{align*} is, when \begin{align*}x =5\end{align*}.

One way, is to use the equation \begin{align*}y = 2x\end{align*} which models this data. Then, you just plug in 5 for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} y &=& 2x \\ y &=& 2(5) \\ y &=& 10 \end{array}\end{align*}

The answer is the total cost of 5 pounds of tomatoes is 10 dollars.

The other way to solve part c, is to complete the pattern (which you are assuming continues) in the table. The pattern is that if \begin{align*}x\end{align*} increases by 1, \begin{align*}y\end{align*} increases by 2.

The completed table is shown below.

Number of Pounds Purchased \begin{align*}(x)\end{align*}

Total Cost in Dollars \begin{align*}(y)\end{align*}
1 2
2 4
3 6
4 8
5 10

Looking at the table, you can see that the answer is the total cost of 5 pounds of tomatoes is 10 dollars.

Examples

Example 1

Earlier, you were given a problem about Samuel’s tern.

The tern dives into the ocean to catch fish. On one such decent, the bird starts from 19 feet in the air and dives down to the surface of the ocean. Its descent can be modeled by the equation below.

\begin{align*}h=19-16.1t^2\end{align*}

In this equation, \begin{align*}t\end{align*} is in seconds and \begin{align*}h\end{align*} is in feet.

Can you make a table from this equation?

First, start when \begin{align*}t=0\end{align*} and then go by small increments of time. Use increments of .2 and then round answers to the nearest tenth. This descent happens quickly.

Height from ocean in feet \begin{align*}(h)\end{align*}  Time in seconds \begin{align*}(t)\end{align*}
0 19
.2 18.4
.4 16.4
.6 13.2
.8 8.7
1 2.9

Next, can you graph this data and connect the points?

The graph below represents this data.

License: CC BY-NC 3.0

For this situation, it makes sense to only look at values when \begin{align*}t>0\end{align*} and \begin{align*}h>0\end{align*}.

Finally, determine if this graph is linear or nonlinear.

Since, a straight line does not go through the points from the table, the graph is nonlinear. You may also notice that for a constant change in \begin{align*}x\end{align*} of .2, \begin{align*}y\end{align*} does not have a constant rate of change.

Example 2

Is the following true or false?

If a function is linear, then, for each constant increase in \begin{align*}x\end{align*}, there is a constant increase in \begin{align*}y\end{align*}. That is, every time \begin{align*}x\end{align*} increases by a constant number, \begin{align*}y\end{align*} will increase by a constant number.

True. Plot points where, every time \begin{align*}x\end{align*} increases by a constant number, \begin{align*}y\end{align*} increases by a constant number. For example, try plotting points where every time \begin{align*}x\end{align*} increases by 3, \begin{align*}y\end{align*} increases by 5. You can connect these points with a straight line, so it is a linear function.

Solve for \begin{align*}y\end{align*} using the given \begin{align*}x\end{align*}, using the equation \begin{align*}y = 2x\end{align*}.

Example 3

When \begin{align*}x = 4 \end{align*} 

\begin{align*}\begin{array}{rcl} y &=& 2x \\ y &=& 2(4) \\ y &=& 8 \end{array}\end{align*}

The answer is \begin{align*}y = 8\end{align*}.

Example 4

When \begin{align*}x = 4.5\end{align*}

\begin{align*}\begin{array}{rcl} y &=& 2x \\ y &=& 2(4.5) \\ y &=& 9 \end{array}\end{align*}

The answer is \begin{align*}y = 9\end{align*}.

Example 5

When \begin{align*}x = 5.5\end{align*}

\begin{align*}\begin{array}{rcl} y &=& 2x \\ y &=& 2(5.5) \\ y &=& 11 \end{array}\end{align*}

The answer is \begin{align*}y = 11\end{align*}.

Review

Look at the information in each table and determine whether the data could be from a linear function or not.

  1.   
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
0 2
1 3
2 5
4 4
  1.   
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
1 3
2 5
3 7
4 9
  1.   
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
2 6
3 9
5 15
6 18
  1.   
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
2 3
3 4
6 7
8 9
  1.   
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
8 4
6 12
2 8
0 0
  1.    
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
0 3
1 4
2 5
6 9
  1.   
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
5 11
4 9
3 7
2 5
  1.   
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
1 7
3 4
2 9
5 8
  1.   
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
1 3
2 6
4 12
6 18
  1.     
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
4 2
5 3
6 5
7 1

The table below 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 ?
  1. Assuming the pattern continues in the table above, write an equation to describe the relationship between \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
  2. 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.

License: CC BY-NC 3.0
 

  1. Assuming the pattern continues, 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. The equation \begin{align*}y = 10 - 2x\end{align*} represents the relationship between \begin{align*}x\end{align*}, the number of bus rides Franklin takes and \begin{align*}y\end{align*}, the number of dollars that are left on his card.

  1. 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, and 3 bus rides.
  2. Create a graph that represents the relationship between the total number of bus rides Franklin takes, \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.

License: CC BY-NC 3.0

  1. If Franklin takes a total of 4 bus rides, how many dollars will be left on his bus card?

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.19.

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Vocabulary

Function

A function is a relation where there is only one output for every input. In other words, for every value of x, there is only one value for y.

Function Rule

A function rule describes how to convert an input value (x) into an output value (y) for a given function. An example of a function rule is f(x) = x^2 + 3.

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

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0
  5. [5]^ License: CC BY-NC 3.0

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