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Section 2.6 Distinguishing Proportional Relationships from Other Linear Relationships

Subsection 2.6.1 Overview

This section focuses on how proportional linear relationships, which we have focused on previously, differ from other types of linear relationships.

Student Page 2.6.2 Giving the Tortoise A Head Start

So far, we have seen several examples of proportional linear relationships. In each case, the graph modeling the relationship went through the origin. We return to The Hare and the Tortoise to consider other possibilities. Giving the Tortoise A Head Start introduces other linear relationships.

The hare challenged the tortoise to another race. He gave the tortoise a head start to make the race seem fair. They used the same 1000-meter course and started the race together.

Figure 2.6.2.1. Empty Axes for Comparing Tortoise and Hare Races

1.

(a)

The tortoise started the race 500 meters from the Starting Line. As in the original race, she plodded at a rate of 20 meters per minute. On Figure 2.6.2.1, draw a graph of the tortoise's distance from the starting line over the time it took her to complete the race.

(b)

The hare ran at a steady rate of 250 meters per minute throughout the race. He stopped only after he crossed the Finish Line. Also on Figure 2.6.2.1, draw a graph of the hare's distance from the starting line over the time it took him to complete the race.

(c)

Who won the race? How do you know?

(d)

Suppose the tortoise started the race 100 meters from the Finish Line. Who would win the race? Explain. Draw the graph on Figure 2.6.2.1.

2.

Complete all three columns in Table 2.6.2.2 to compare the tortoise's distance from the Starting Line at the same time for these races. For each race, use a different color to highlight the time at which the tortoise reached the Finish Line. Use the same color to highlight the corresponding graph in Figure 2.6.2.3. Recall: In the original race, the tortoise started at the Starting Line.

Table 2.6.2.2. Distance and Time of the Tortoise's Travels
  The tortoise's distance in meters from the Starting Line
Time elapsed
in minutes
Original Race Race in Task 2.6.2.1.a Race in Task 2.6.2.1.d
0 0    
1 20    
2      
3      
4      
5      
10      
20      
30      
40      
50      

3.

Graph the tortoise's races from Table 2.6.2.2 on the same coordinate plane in Figure 2.6.2.3. How are the graphs related?

Figure 2.6.2.3. Time and Distance for Various Tortoise Races

4.

Study the table and the graphs.

(a)

For each graph and corresponding data set, find an equation to model the graph and data.

(b)

What do the equations have in common? Explain.

(c)

What is different in each of the equations? What accounts for the differences?

(d)

How does Tortoise's head start show up in each equation? Why does this make sense?

5.

You have studied the idea of ‚Äúhead start‚ÄĚ in linear relationships in previous mathematics classes. What is the mathematical name for the ‚Äúhead start‚ÄĚ value in a linear equation?

With your group, discuss what new information about linear relationships is gained from the student page, Giving the Tortoise A Head Start. Outline what you have learned about linear relationships through the progression of activities: The Hare and the Tortoise, Trodding Tortoise, Hopping Hare, and Giving the Tortoise A Head Start. Ask any questions you have remaining.

Student Page 2.6.3 Finding Equations from Tables and Interpreting Slope and y-Intercept

Sometimes students find it difficult to determine an equation from a table, a graph, or a story. These three representations are highly linked. If you can find a table from a graph or a story, you can always find an equation from the patterns in the table. The student page, Finding Equations from Tables and Interpreting Slope and y-Intercept, gives you a few tools to help you find an equation from a table. Though initially, this work might seem time-consuming, complete it as directed. It is important not to simplify too soon so that you can see how patterns are forming. You will find short-cuts soon enough.

Work with your group to complete the student page, Finding Equations from Tables and Interpreting Slope and y-Intercept. Pay close attention to how you can find equations from tables. Be ready to answer these questions:

  • How are recursive and explicit equations related?

  • How does this activity help you distinguish proportional linear relationships from other linear relationships?

1.

(a)

What patterns do you see in Table 2.6.3.1? Table 2.6.3.2? (Don't complete the tables yet.)

Table 2.6.3.1. Table A
\(x\) \(y\) Another way to write \(y\)
0 0 0
1 3 \(0 + 3\)
2 6 \(0 + 3 + 3\)
3 9  
4 12  
5    
6    
27    
108    
Table 2.6.3.2. Table B
\(x\) \(y\) Another way to write \(y\)
0 5 5
1 8 \(5 + 3\)
2 11  
3 14  
4 17  
5    
6    
27    
108    

2.

Graph the data in both tables on the same pair of axes. What do you notice? Does your observation seem reasonable? Explain.

3.

(a)

In the column labeled, ‚ÄúAnother way to write \(y\)‚ÄĚ, show how the \(y\)-values in the table build from the \(y\)-value when \(x = 0\text{.}\) This is the starting \(y\)-value. Do not simplify at any step. Continue extending the table to find the \(y\)-values for \(x = 5\) and \(x = 6\text{.}\)

(c)

A recursive pattern is a rule that tells you the starting number of a pattern and how the pattern continues. It is likely that recursive patterns are the first thing you notice when working with tables. For both Table 2.6.3.1 and Table 2.6.3.2, write a recursive rule to find the \(y\)-values in the table. Don't forget to list a starting \(y\)-value and what you have to do to get successive \(y\)-values in the table.

Table 2.6.3.1:

Table 2.6.3.2:

4.

An explicit equation is a rule that allows you to find a value for \(y\) directly from the corresponding value for \(x\text{.}\) In the recursive pattern, you found that you added the same number each time. Let's call this number a constant. In the table, for a particular value of \(x\text{,}\) how many times did you add the constant to get the value of \(y\) for that value of \(x\text{?}\) Use this information to write explicit equations for Table 2.6.3.1 and Table 2.6.3.2. Use the equations to complete the tables.

Table 2.6.3.1:

Table 2.6.3.2:

5.

How does each explicit equation relate to each recursive equation?

6.

Repeat Student Page Exercise¬†2.6.3.1‚Äď2.6.3.4 for Table¬†2.6.3.3 and Table¬†2.6.3.4.

Table 2.6.3.3. Table C
\(x\) \(y\) Another way to write \(y\)
0 9  
1 7  
2 5  
3 3  
4 1  
5 \(-1\)  
6 \(-3\)  
Table 2.6.3.4. Table D
\(x\) \(y\) Another way to write \(y\)
1 6  
2 8  
3 10  
4 12  
5 14  
6    
7    

Recursive rule:

Explicit equation:

7.

Are you tired of filling out the column, ‚ÄúAnother way to write \(y\)‚ÄĚ? Now that you've experienced the constant growth of a linear relationship, you can simplify the process. Study Table¬†2.6.3.5. Notice how the process below simplifies the work you did in Student Page Exercise¬†2.6.3.1 and Student Page Exercise¬†2.6.4.6. In this process you find the difference between two consecutive \(y\)-values instead of writing each \(y\)-value in terms of the starting \(y\)-value. For Table¬†2.6.3.5:

(a)

How many times do you add 4 to the starting \(y\)-value to get the \(y\)-value when \(x = 3\text{?}\)

(b)

How many times do you add 4 to the starting \(y\)-value to get the \(y\)-value when \(x = 6\text{?}\)

(c)

Find an explicit equation to fit the table. How do you know your equation is correct?

(d)

Use the same process to analyze Table 2.6.3.6. Find an explicit equation to fit the table.

Table 2.6.3.5. Table E
\(x\) \(y\) Change in \(y\)
0 7 Starting value
1 11 4
2 15 4
3 19 4
4 23 4
5 27 4
6 31 4
Table 2.6.3.6. Table F
\(x\) \(y\) Change in \(y\)
0 15  
1 12  
2 9  
3 6  
4 3  
5 0  
6 \(-3\)  

8.

There are two important numbers in a linear equation.

(a)

One is the slope. How can you find the slope from a table? How does it show up in the equation?

(b)

The other number is called the \(y\)-intercept. What is a \(y\)-intercept?

(c)

How can you find the \(y\)-intercept from a table? How does it show up in the equation?

Student Page 2.6.4 Finding Equations from Graphs and Contexts, Interpreting Slope and y-Intercept

1.

Compare the graphs in Figure 2.6.4.1. Which ones are related to each other? How are they related? (Note: The green line is Graph A, blue is Graph B, black is Graph C, and red is Graph D)

Figure 2.6.4.1. Four Linear Plots

2.

(a)

Complete a table for each graph. Use Table 2.6.4.2, Table 2.6.4.3, Table 2.6.4.4, and Table 2.6.4.5.

Table 2.6.4.2. Graph A (Green)
\(x\) \(y\)
0    
1    
2    
3    
4    
5    
6    
Table 2.6.4.3. Graph B (Blue)
\(x\) \(y\)
0    
1    
2    
3    
4    
5    
6    
Table 2.6.4.4. Graph C (Black)
\(x\) \(y\)
0    
1    
2    
3    
4    
5    
6    
Table 2.6.4.5. Graph D (Red)
\(x\) \(y\)
0    
1    
2    
3    
4    
5    
6    
(b)

Determine a recursive pattern for each table.

(c)

Use the recursive rule to find an explicit equation for each graph. Write the recursive pattern and the equation under each table.

3.

(b)

How can you find the slope of a graph from a table?

(c)

How can you find the slope of a graph from an equation?

(d)

How can you find the slope directly from a graph? Show how the slope arises from the graph in each case.

4.

(a)

What is the \(y\)-intercept?

(b)

How can you find the \(y\)-intercept from a table?

(c)

How can you find the \(y\)-intercept from an equation?

(d)

How can you find the \(y\)-intercept from a graph?

5.

Consider the contexts that follow.

Context A

Raffle tickets are $0.50 each. How much will you pay for \(T\) tickets?

Context B

You have $6 to buy ride tickets at a fair. Each ticket costs $1. How much money do you have after you purchase \(T\) tickets?

Context C

It costs $2 to get into a dance. At the dance you can buy glasses of punch for $0.50 each. How much will you spend for the dance and punch after buying \(G\) glasses of punch?

Context D

The Rowing Team is selling cookies for $1 each. A friend is selling the cookies and allows you to pay later for cookies you buy beyond the $3 you have in your pocket. How much will you pay your friend if you buy \(C\) cookies? (Negative dollars are dollars you will pay your friend after the cookie sale.)

(b)

Graph the data in each table.

7.

Match the contexts in Student Page Exercise 2.6.4.6 to the graphs in Student Page Exercise 2.6.4.1. How do you know you are right?

\begin{align*} \amp \text{Graph A} \amp \qquad \amp \text{Context A}\\ \amp \text{Graph B} \amp \qquad \amp \text{Context B}\\ \amp \text{Graph C} \amp \qquad \amp \text{Context C}\\ \amp \text{Graph D} \amp \qquad \amp \text{Context D} \end{align*}

8.

Some students say that you can find the slope from a table by dividing the value of \(y\) by the value of \(x\) for a single data point.

(a)

Does this method ever work? If so, for which of the examples above will this method work? Why does the method work?

(b)

Try this method for finding slope on Table 2.6.4.3. Does the method work in this case? Why or why not?

(c)

For which of the examples above will this method for finding slope fail? Why does it fail?

Homework 2.6.5 Homework

2.

Look for uses of linear equations in your daily life and write a problem similar to those in Student Page Exercise 2.6.4.5. The problem can come from a newspaper, some other media source, through a table, graph, or story. It must be specific enough for others to determine an equation from your example. Find at least 3 examples, one of which includes the point (0, 0). Write your examples on index cards to pass around in class.

3.

For each of the following tasks, complete parts Item¬†2.6.5.3.c‚Äď2.6.5.3.f Complete Item¬†2.6.5.3.a and Item¬†2.6.5.3.b for the problems to which they pertain.

  1. Fill-in the first 5 missing values in the tables (Task 2.6.5.3.a, Task 2.6.5.3.b, Task 2.6.5.3.e, Task 2.6.5.3.f).

  2. Write a story to go with the two completed tables provided.

  3. Identify the slope. State the meaning of the slope in context.

  4. Determine an equation. Make sure your equation fits the data.

  5. Tell how you know your equation is correct in terms of the story or graph.

  6. Determine and write the last value in each table.

(a)
(i) Story.

Nasredinne drinks 64 ounces of water each day. What is the total amount of water, \(y\text{,}\) he has consumed by the end of \(x\) days?

(ii) Table.
\(x\) \(y\)
1  
2  
3  
4  
5  
28  
(iii) Slope, Equation, and Explanation.

Answer:

(b)
(i) Story.

Jessa jumps rope for \(\frac{3}{4}\) hours per day. What is the total number of hours, \(y\text{,}\) she has jumped rope by the end of \(x\) days?

(ii) Table.
\(x\) \(y\)
1  
2  
3  
4  
5  
31  
(iii) Slope, Equation, and Explanation.

Answer:

(c)
(i) Story.

Answer:

(ii) Table.
\(x\) \(y\)
1 1.2
2 2.4
3 3.6
4 4.8
5 6.0
17  
(iii) Slope, Equation, and Explanation.

Answer:

(d)
(i) Story.

Answer:

(ii) Table.
\(x\) \(y\)
1 7
2 14
3 21
4 28
5 35
43  
(iii) Slope, Equation, and Explanation.

Answer:

(e)
(i) Graph.
(ii) Table.

Do not estimate. Use grid points to fill in the first 5 values of the table.

\(x\) \(y\)
   
   
   
   
   
8  
(iii) Slope, Equation, and Explanation.

Answer:

(f)
(i) Graph.
(ii) Table.

Do not estimate. Use grid points to fill in the first 5 values of the table.

\(x\) \(y\)
   
   
   
   
   
32  
(iii) Slope, Equation, and Explanation.

Answer:

4.

Find two related data sets (or create them from real contexts such as a menu). One data set must be a proportional linear relationship. The other data set must be linear but not a proportional linear relationship. Answer the following questions for each data set:

(a)

Present your scenario. Complete a table to fit the scenario.

(b)

Create a graph using an electronic tool. Label the horizontal (\(x\)) axis and the vertical (\(y\)) axis. Plot the data from Task 2.6.5.4.a. Connect the points in your graph. Print the graph.

(c)

Describe the shape and important features of the graph you created.

(d)

Is the relationship between \(x\) and \(y\) linear? Why or why not?

(e)

How can you tell from the graph whether or not the linear function is proportional? Explain.

(f)

Write an equation that gives \(y\) in terms of \(x\text{.}\)

(g)

Compare the data you found for the proportional linear function with the data you found for the linear function. What accounts for the differences between the data sets?