Determining Linear Rules from Data and Graphs

TL;DR

You can figure out the mathematical rule that connects two variables (like x and y) by looking at a table of values or a graph. If the relationship forms a straight line, it's a linear rule, often written as an equation. By finding patterns in the numbers or points, you can construct this rule to describe the relationship.

1. The Mental Model

Imagine you have some pairs of numbers, like (x, y). If these pairs consistently change in a predictable way that forms a straight line when plotted, there's a simple mathematical rule linking them. Your goal is to uncover that hidden rule from the data you have.

2. The Core Material

A rule is an equation that shows how two or more variables are connected. For linear relationships, this rule will give you a straight line when graphed.

Understanding the Relationship Between Rules, Tables, and Graphs

Think of these three as different ways to show the same relationship:

• Rule (Equation): This is the mathematical formula, like y = 2x - 1.
• Table of Values: This lists several x values and their corresponding y values based on the rule.
• Graph: This is a visual representation of the (x, y) pairs plotted on a coordinate plane, forming a line for linear rules.

You'll see how a point (x, y) lies on the graph of an equation if substituting its x and y values into the equation makes the equation true.

graph TD
    A["Rule (e.g., y = 2x - 1)"] --> B["Construct Table of Values"]
    B --> C["Plot Points on Graph"]
    C --> D["Observe Linear Relationship"]
    D --> E["Derive Rule from Graph (if starting from graph)"]
    E --> A
    B --> E

Constructing a Table from a Rule

To create a table of values from a rule (e.g., y = 2x - 1):
1. Choose several x values (it's often good to pick a mix of negative, zero, and positive numbers).
2. Substitute each x value into the rule to calculate the corresponding y value.
3. Record the (x, y) pairs in a table.

Example: For the rule y = 2x - 1
| x | Calculation (2x - 1) | y | (x, y) |
|---|---|---|---|
| -2 | 2(-2) - 1 = -5 | -5 | (-2, -5) |
| -1 | 2(-1) - 1 = -3 | -3 | (-1, -3) |
| 0 | 2(0) - 1 = -1 | -1 | (0, -1) |
| 1 | 2(1) - 1 = 1 | 1 | (1, 1) |
| 2 | 2(2) - 1 = 3 | 3 | (2, 3) |

Plotting a Graph from a Table or Rule

Once you have (x, y) pairs from your table:
1. Draw a number plane (Cartesian plane) with x and y axes.
2. Plot each (x, y) pair as a point on the plane.
3. If the relationship is linear, connect the points with a straight line.

Finding the Rule from a Table of Values

This is where you reverse the process! When given a table of values, look for patterns:
1. Check the change in y for each unit change in x: Is y increasing or decreasing by the same amount each time x increases by 1? This constant change is important.
2. Look for the y-value when x is 0: This will often be a key part of your rule.

For simple linear rules in the form y = mx + c:
* m represents the constant change in y for every unit change in x.
* c represents the y-value when x = 0.

Finding the Rule from a Linear Graph

If you have a linear graph and know the coordinates of at least two points (especially integer values of x and y):
1. Identify two clear points on the line, say (x1, y1) and (x2, y2).
2. Determine the "steepness" or gradient (m): This is the change in y divided by the change in x between your two points: m = (y2 - y1) / (x2 - x1).
3. Find the y-intercept (c): This is the y-value where the line crosses the y-axis (where x = 0). You might be able to read this directly from the graph.
4. Write the rule: Once you have m and c, you can write the rule as y = mx + c.

3. Worked Example

Let's find the rule from this table of values:

1. Check the change in y for each unit change in x:

   • From x = -1 to x = 0 (change in x is +1), y goes from 5 to 3 (change in y is -2).
   • From x = 0 to x = 1 (change in x is +1), y goes from 3 to 1 (change in y is -2).
   • From x = 1 to x = 2 (change in x is +1), y goes from 1 to -1 (change in y is -2).
     Since the change in y is consistently -2 for every +1 change in x, our m (gradient) is -2. So, the rule starts with y = -2x.

2. Look for the y-value when x is 0:
   From the table, when x = 0, y = 3. This means our c (y-intercept) is 3.

3. Combine to form the rule:
   The rule is y = -2x + 3.

Let's quickly check this:
If x = -1, y = -2(-1) + 3 = 2 + 3 = 5. (Matches table)
If x = 2, y = -2(2) + 3 = -4 + 3 = -1. (Matches table)
It works!

4. Key Takeaways

• A linear rule describes a straight-line relationship between variables, usually x and y.
• You can move between a rule (equation), a table of values, and a graph, as they all represent the same relationship.
• To find a rule from a table, look for the constant change in y for each unit change in x (this is your gradient m).
• Also, find the y-value when x is 0; this is your y-intercept c.
• The general form of a linear rule you're looking for is y = mx + c.
• You can verify a point (x, y) is on a graph or follows a rule by substituting its values into the equation.

Common Mistakes to Avoid:
- Not checking for a constant change in y for a unit change in x – if it's not constant, the relationship isn't linear.
- Confusing the x-intercept with the y-intercept when reading from a graph or table.
- Calculating the gradient m incorrectly (it's change in y / change in x, not the other way around).
- Not testing your derived rule with a few points from the original data to confirm its accuracy.

5. Now Try It

For the x-coordinates from -2 to 2, construct a table and draw a graph for the rule y = 3x - 1. Then, imagine you only had the table and graph you just made; try to work backward to "re-discover" the rule y = 3x - 1 by identifying the constant change in y and the y-value when x is 0.

What success looks like:
You'll have a table with x values from -2 to 2 and their corresponding y values. You'll have a graph with these points plotted and connected by a straight line. Finally, you'll be able to explain how you found that m = 3 and c = -1 from your table/graph to correctly state the rule y = 3x - 1.