Repeated Change as a Rule

1. Hook: Why This Matters

Some tables look like many separate rows, but a steady habit may be hiding inside them. For example, every time x increases by 1, y might increase by 3.

When that habit repeats, we do not have to memorize the whole table. We can write a rule.

2. Intuition: See Repeated Change as a Table's Habit

Two things matter most: the starting value and the repeated increase or decrease.

3. Formal Definition: The Rule

A basic linear rule can be written as

• b is the starting value when x = 0
• m is how much y changes when x increases by 1

If b = 2 and m = 3, then

4. Interactive Exploration: Change the Rule and Watch the Graph

Slope m3y-intercept b2Choose x1

Current formula

$$y=3x+2$$

When you choose x = 1

$$y=3(1)+2=5$$

Observation

Right now, m = 3, so the line rises as you move to the right. When x = 1, y = 5.

Guiding questions:

• When m = 3, how much does y increase each time x increases by 1?
• When you change b, how does the starting value in the table change?
• If m is negative, what story would the table tell?

5. Step-by-step Example

Example

This table starts at 2 and increases by 3 each step.

Write a rule for the table.

Step-by-step Thinking

1. Look at the row x = 0: y = 2, so the starting value is b = 2.
2. Find the change in y: from 2 to 5 is an increase of 3.
3. Check the next row: from 5 to 8 is also an increase of 3.
4. So the repeated change is m = 3.
5. Write the rule as y = 3x + 2.

6. Common Mistakes

Common mistake

Do not treat the first visible y value as the starting value unless that row has x = 0. If the table starts later, you may need to reason backward.

Common mistake

Do not decide a table is linear from only two rows when more rows are given. Check that the change in y stays steady over the part you are studying.

7. Mini Exercise

Try this

Does this table have repeated change? If it does, what is the rule?

x	y
0	-1
1	1
2	3
3	5

1. What is the starting value?
2. How much does y change when x increases by 1?
3. Write the rule in the form y = mx + b.

8. Summary

• Repeated change means a value changes by the same amount each step.
• In y = mx + b, m is the repeated change per 1 unit of x.
• b is the starting value when x = 0.
• A table with steady repeated change often connects to a straight-line graph.

9. Related Lessons

• Plotting Points From a Table
• Understanding Linear Functions Through Slope
• Slope as a Rate of Change