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# Linear and Non-Linear Patterns of Change

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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 $= 85 \ ft$

Speed $= 55 \ mph$

Kingda Ka Roller Coaster

Height $= 456 \ feet$

Speed $= 128 \ mph$

Top Thrill Dragster Roller Coaster

Height $= 420 \ ft.$

Speed $= 120 \ mph$

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 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.

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 $(x)$ Total Cost in Dollars $(y)$
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, $x$ , and the total cost, $y$ .

c. Determine the cost of buying 5 pounds of tomatoes at the farmer's market.

Consider part $a$ first.

Use guess and check to determine how to each pair of values is related.

For example, notice that each $y-$ value is greater than its corresponding $x-$ value. So, the rule will involve either addition or multiplication.

Since each $y-$ value in the table is 2 more than its previous $y-$ value, the rule may involve multiplying by 2.

Look for a rule that involves multiplying by 2.

Consider the ordered pair (1, 2).

$1 \times 2=2$ , so the rule could be to multiply each $x-$ value by 2 to find its corresponding $y-$ value. Check to see if that rule works for the other pairs of values in the table.

Consider the ordered pair (2, 4).

$2 \times 2=4$ , so the rule works for that ordered pair.

Consider the ordered pair (3, 6).

$3 \times 2=6$ , so the rule works for that ordered pair.

So, the rule for this function table is: multiply each $x-$ value by 2 to find its corresponding $y-$ value.

Now that we have a rule in words, let's write an equation to show the same relationship. Remember, to find each $y-$ value, you must multiply each $x-$ value by 2. So, the equation would be $y=2x$

Next, consider part $b$ .

First, let's consider how the graph for this function should look. We should use the horizontal axis to show the number of pounds, $x$ .

We should use the vertical axis to show the total cost, $y$ , in dollars.

Consider how to number the axes. The $x-$ values represent the number of pounds of tomatoes purchased. Since no one can buy a negative number of pounds of tomatoes, the graph should only show $x-$ values that are greater than or equal to 0. So, we can number each axis starting from 0. The greatest $x-$ value we must show is 3. The greatest $y-$ value we must show is 6. So, the numbers on the vertical axis must go up to at least 6. Let's allow some room for the graph to be extended and number each axis from 0 to 10.

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, $x$ , and the total cost, $y$ .

Finally, consider part $c$ .

One strategy for determining the total cost, $y$ , of buying 5 pounds of tomatoes is to use the equation. We can substitute 5 for $x$ and then solve to find the value of $y$ .

$y &= 2x\\y &= 2 \times 5\\y &= 10$

When $x = 5, y = 10$ . So, it would cost 10 dollars to buy 5 pounds of tomatoes.

Now it's your turn to try a few. Find $y$ use the equation $y = 2x$ .

#### Example A

When $x = 4$

Solution: $y = 8$

#### Example B

When $x = 4.5$

Solution: $y = 9$

#### Example C

When $x = 5.5$

Solution: $y = 11$

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 $= 85 \ ft$

Speed $= 55 \ mph$

Kingda Ka Roller Coaster

Height $= 456 \ feet$

Speed $= 128 \ mph$

Top Thrill Dragster Roller Coaster

Height $= 420 \ ft.$

Speed $= 120 \ mph$

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.

$H$ $S$
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

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 $x$ and $y$ value must increase or decrease at the same interval.

True. This graph will be straight line. Otherwise, it is a non - linear function.

### Practice

Directions: Look at each table and determine whether each pattern of change is linear or non-linear.

1.

$x$ $y$
0 2
1 3
2 5
4 4

2.

$x$ $y$
1 3
2 5
3 7
4 9

3.

$x$ $y$
2 6
3 9
5 15
6 18

4.

$x$ $y$
2 3
3 4
6 7
8 9

5.

$x$ $y$
8 4
6 12
2 8
0 0

6.

$x$ $y$
0 3
1 4
2 5
6 9

7.

$x$ $y$
5 11
4 9
3 7
2 5

8.

$x$ $y$
1 7
3 4
2 9
5 8

9.

$x$ $y$
1 3
2 6
4 12
6 18

10.

$x$ $y$
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 $(x)$ Total Cost in Dollars $(y)$
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, $x$ , and the total cost, $y$ . 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 $x$ , the number of times he uses his card to ride the bus and $y$ , the number of dollars that are left on his card:

$y=10-2x$ .

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, $x$ , and the number of dollars left on the card, $y$ . 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?