1. What is Slope?

Slope measures the steepness and direction of a line. It's a fundamental concept in algebra and calculus that describes how much a line rises or falls as it moves horizontally.

2. How to Calculate Slope

The slope (m) between two points (x₁,y₁) and (x₂,y₂) is calculated using:

Where:

• \( (x_1, y_1) \) — Coordinates of the first point
• \( (x_2, y_2) \) — Coordinates of the second point

Explanation: The numerator represents the vertical change (rise), while the denominator represents the horizontal change (run).

3. Understanding the Results

Details:

• Positive slope: Line rises from left to right
• Negative slope: Line falls from left to right
• Zero slope: Horizontal line
• Undefined slope: Vertical line (when x₁ = x₂)

4. Practical Applications

Examples: Slope is used in physics (velocity), engineering (gradients), economics (rate of change), and computer graphics (line drawing algorithms).

5. Frequently Asked Questions (FAQ)

Q1: What if my points are the same?
A: If (x₁,y₁) = (x₂,y₂), the slope is undefined as you can't determine a single line through one point.

Q2: Does the order of points matter?
A: No, (y₂-y₁)/(x₂-x₁) gives the same result as (y₁-y₂)/(x₁-x₂).

Q3: How precise should my coordinates be?
A: Use at least 2 decimal places for accurate results in most applications.

Q4: What's the slope of a 45° line?
A: A line at exactly 45° has a slope of 1 (rises 1 unit for every 1 unit run).

Q5: Can slope be used for curves?
A: For curves, slope varies at each point and is calculated using derivatives in calculus.