Lesson 6.4
Writing Equations from Two Points
This is the "Boss Fight" of the chapter. You have to use every skill you've learned so far: finding slope, solving for b, and writing the equation.
Introduction
Sometimes you don't know the rate () OR the start (). You just have two pieces of data. "In year 1, profits were $50k. In year 3, profit were $90k." From just that, you can build the entire model.
Past Knowledge
Lesson 5.3 (Slope Formula). You MUST be able to calculate .
Today's Goal
Combine steps: Find first, THEN find .
Future Success
This is the most common way real-world models are built from raw data.
Key Concepts
The Full Process
- Calculate Slope ().
- Pick ONE point.
It doesn't matter which one. Pick the easier numbers (positives, smaller numbers).
- Solve for .
Plug into .
- Write the Equation.
Worked Examples
Example 1: Positive Slope
BasicFind the line passing through and .
Step 1: Find Slope
Step 2: Solve for b
Pick .
Example 2: Negative Slope
IntermediateFind the line passing through and .
Step 1: Find Slope
Step 2: Solve for b
Pick .
Add 8 to both sides.
Example 3: Horizontal Line
AdvancedFind the line passing through and .
Step 1: Find Slope
Step 2: Solve for b
Remember HOY (Horizontal, Zero slope, y=)
Common Pitfalls
Arithmetic Errors
Slope formula involves subtraction. If you have negative numbers, be careful! becomes . Double check your signs.
Order Matters (Sort of)
If you start with on top, you MUST start with on bottom. Don't criss-cross your points.
Real-Life Applications
Business Growth:
- Year 1: 100 customers. Year 5: 500 customers.
- Slope: customers per year.
- You can now predict Year 10 just from those two points!
Practice Quiz
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