Lesson 5.1

Types of Slope

Is it an uphill climb, a downhill slide, a flat road, or a sheer cliff? Slope isn't just a number; it's a direction.

Introduction

In algebra, Slope measures the steepness and direction of a line. Before we calculate it with numbers, we need to recognize it with our eyes. Think of reading a book: always move from Left to Right.

Past Knowledge

Lesson 4.6 (Qualitative Graphs). You already know "Increasing" vs "Decreasing".

Today's Goal

Identify the four types of slope: Positive, Negative, Zero, and Undefined.

Future Success

In Calculus, "Slope" becomes the "Derivative". Recognizing zero slope is key to finding maximums and minimums.

Key Concepts

The Four Faces of Slope

Always Read from Left to Right →

Positive

Going UP hill.

Like an airplane taking off.

Negative

Going DOWN hill.

Like a slide or crash.

Zero

Horizontal Flat.

Walking on flat ground. Easy.

Undefined

Vertical Cliff.

Impossible to walk. Falling.

Worked Examples

Example 1: Identify the Slope

Basic

Classify the slope of the line shown.

Step 1: Left to Right

Put your finger on the left side of the line.

Step 2: Trace

Move your finger to the right. Does it go up or down?

It goes UP.

Positive Slope

Example 2: A Tricky Case

Intermediate

Classify the slope.

Step 1: Analyze Flatness

The line is perfectly horizontal.

Step 2: Can you walk on it?

Yes, it's like a floor. It takes ZERO effort to walk on flat ground.

Zero Slope

Example 3: The Vertical Line

Advanced

Classify the slope defined by .

Step 1: Analyze direction

It goes straight up and down. It's a wall.

Step 2: Can you walk on it?

No. You would fall. Calculations divide by zero.

Undefined Slope

Common Pitfalls

Zero vs. Undefined

Zero is a number (0). It exists. You can have \$0 in your bank account.

Undefined means "does not exist" or "infinity". It's a crash. Do not confuse them.

Reading Right to Left

If you read a book backwards, the story makes no sense. If you read a graph from Right to Left, "Uphill" looks like "Downhill". Always go Left → Right.

Real-Life Applications

Roof Pitch: Builders need to know the slope of a roof so snow and rain slide off (Positive/Negative slope). A flat roof (Zero slope) collects water and leaks. A wall (Undefined slope) is just... a wall.