Lesson 6.5

Point-Slope Form

Slope-Intercept form () is great, but it has a weakness: it requires . Point-Slope form works with ANY point.

Introduction

Imagine you want to give directions to your house.
Slope-Intercept says: "Start at the city center (), then go north."
Point-Slope says: "Start where you are right now , then go north."

Past Knowledge

Lesson 5.3 (Slope). This new formula is actually just the slope formula rearranged!

Today's Goal

Master the form .

Future Success

In Calculus, this is the main way we write equations for tangent lines.

Key Concepts

Where does it come from?

Start with the slope formula:

Multiply both sides by the denominator $(x-x_1)$:

That's it! It's just the slope formula without the fraction.

Anatomy of the Formula

Y-Coordinate

Slope

X-Coordinate

Vital Rule: The sign in the formula is MINUS.

So if you see , the coordinate is POSITIVE 3. If you see , it means , so the coordinate is NEGATIVE 5.

Worked Examples

Example 1: Identify the Parts

Basic

Find the slope and point from:

Step 1: Identify m

It's the number in front of the parenthesis.

Step 2: Identify Point

means .

Point:

Example 2: Writing the Equation

Intermediate

Write an equation for the line through with slope .

Step 1: Plug into Formula

Step 2: Simplify Signs

Minus a negative becomes plus.

Example 3: Graphing

Advanced

Graph .

Step 1: Extract Point

means .

Start at:

Step 2: Use Slope

. Go up 1, right 2.

Common Pitfalls

Sign Confusion

Students see and think the coordinate is . NO. The form has a built-in minus. "Take the opposite sign."

Distributing Too Early

Sometimes you don't need to simplify! If the question asks for "Point-Slope Form", stop at . Don't solve for y unless asked.

Real-Life Applications

Science Experiments:

You usually don't measure the "start" (). You measure things in the middle of the experiment.
"At 5 minutes, temp was 40 degrees." Point: .
Because you have a point that isn't the y-intercept, Point-Slope form is the natural way to write the equation.