Writing Equations in Point-Slope Form
— when you know the slope and anypoint, point-slope form is often the fastest way to write a line's equation.
Introduction
Slope-intercept form requires the y-intercept, which you don't always have. Point-slope form lets you write the equation using any point on the line — no need to solve for b first.
Past Knowledge
Slope-intercept form (3.2.4). Slope formula (3.2.1).
Today's Goal
Write equations in point-slope form and convert between forms.
Future Success
Coordinate proofs and finding point-to-line distance (3.2.6) use this form extensively.
Key Concepts
Point-Slope Form
= slope, = any known point on the line.
When to Use Each Form
| You Know | Best Form |
|---|---|
| Slope + y-intercept | Slope-Intercept () |
| Slope + any point | Point-Slope () |
| Two points | Find slope → use either form |
Interactive Diagram — Desmos Geometry
A line through a known point — see how point-slope form uses any point and the slope to define the line.
Worked Examples
Slope and a Point
Write the equation of the line with slope 4 through .
→
Two Points → Point-Slope → Slope-Intercept
Write the equation through and in both forms.
Slope:
Point-Slope:
Simplify: →
Point-Slope:
Slope-Intercept:
Perpendicular Line Through a Point
Write the equation of the line perpendicular to through .
Perp. slope: (negative reciprocal of )
Point-Slope:
Slope-Intercept:
Common Pitfalls
Sign Errors with Negatives
If the point is , the equation reads , not . Watch the double negative carefully.
Real-Life Applications
Weather Forecasting
Meteorologists use known data points (temperature at specific times) and rate of change to model temperature trends. Point-slope form lets them quickly build a linear model from any two readings.
Practice Quiz
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