Slope

Slope#

Slope of a Line#

Theory#

The slope of a line is a measure of its steepness and direction. It represents the rate of change of the y-coordinate with respect to the x-coordinate.

Given two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, the slope \(m\) is calculated as:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

where \(x_2 \neq x_1\).

Key Properties:

  • 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 \(x_2 = x_1\))

The slope can also be interpreted as:

  • Rise over run: The vertical change divided by the horizontal change

  • Gradient: Another term commonly used for slope

  • Rate of change: In real-world applications

Interactive Visualization: Slope Explorer#

Application#

Examples#

Example 1: Finding the slope between two points

Find the slope of the line passing through the points \(A(2, 3)\) and \(B(6, 11)\).

Solution: Using the slope formula: $\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2\)$

Therefore, the slope is 2, meaning the line rises 2 units for every 1 unit it moves to the right.

Example 2: Determining the type of line

Classify the following lines based on their slopes:

  • Line through \((1, 5)\) and \((4, 5)\)

  • Line through \((3, 2)\) and \((3, 8)\)

  • Line through \((0, 4)\) and \((2, 0)\)

Solution:

  1. For \((1, 5)\) and \((4, 5)\): \(m = \frac{5 - 5}{4 - 1} = \frac{0}{3} = 0\) → Horizontal line

  2. For \((3, 2)\) and \((3, 8)\): \(m = \frac{8 - 2}{3 - 3} = \frac{6}{0}\) → Undefined (Vertical line)

  3. For \((0, 4)\) and \((2, 0)\): \(m = \frac{0 - 4}{2 - 0} = \frac{-4}{2} = -2\) → Negative slope

Example 3: Real-world application

A road rises 15 meters over a horizontal distance of 200 meters. Find the slope and express it as a percentage gradient.

Solution:

  • Slope = \(\frac{\text{rise}}{\text{run}} = \frac{15}{200} = 0.075\)

  • Percentage gradient = \(0.075 \times 100\% = 7.5\%\)

This means the road has a 7.5% gradient.

Multiple Choice Questions#

Sector Specific Questions: Slope Applications#

Key Takeaways#

Important

Essential Slope Concepts:

  1. Slope Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for points \((x_1, y_1)\) and \((x_2, y_2)\)

  2. Types of Slopes:

    • Positive: Line rises left to right

    • Negative: Line falls left to right

    • Zero: Horizontal line

    • Undefined: Vertical line

  3. Perpendicular Lines: Product of slopes = -1

  4. Parallel Lines: Same slope

  5. Real-World Applications: Gradients, rates of change, engineering design

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