Understanding Gradient

Definition of Gradient

• A measure of how steep a line is.
• Positive gradient: Line increases in height.
• Zero gradient: Line is flat.
• Negative gradient: Line decreases in height.

Calculating Gradient

Method 1: Rise per Run

• Determine how much the line rises for each unit it runs across.
• Example:
  • If a line rises 1 unit for each 1 unit it runs, the gradient is 1.
  • If a line rises 0.5 unit for each 1 unit it runs, the gradient is 0.5.

Method 2: Rise over Run Equation

• Formula: Gradient = Rise / Run.
• Rise: Vertical change (change in y).
• Run: Horizontal change (change in x).
• Example:
  • A rise of 0.5 and run of 1 gives a gradient of 0.5.

Method 3: Change in Y over Change in X

• Formula: Gradient = (Change in Y) / (Change in X).
• Equivalent to the rise over run method.
• Example:
  • From x = -4 to x = 2, x changes by 6.
  • From y = -1 to y = 2, y changes by 3.
  • Gradient = 3 / 6 = 0.5.

Examples

Lines and Their Gradients

• Flat Line:
  • Gradient = 0 (no rise).
• Descending Line:
  • Traveling left to right, the line goes down.
  • Example:
    • If it goes down by 2 when it runs across 1, gradient = -2.
    • Using the formula:
      • Change in Y = -6 (from 3 to -3)
      • Change in X = 3 (from -1 to 2)
      • Gradient = -6 / 3 = -2

Summary

• Three methods to calculate gradient: direct observation, rise/run equation, and change in y/change in x equation.
• All methods provide the same result when applied correctly.

Remember: Always consider the line from left to right to correctly determine the gradient sign.