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Student Page 2.6.3 Finding Equations from Tables and Interpreting Slope and y-Intercept
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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, “Another way to write \(y\)”, 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 2.6.3.1–2.6.3.4 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, “Another way to write \(y\)”? 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?