Lesson 6: Gradient Using the Formula

Learning Objectives

By the end of this lesson, students will be able to:

  • Calculate the gradient of a line between two points using the formula
  • Interpret the meaning of positive, negative, zero, and undefined gradients
  • Check gradient calculations for reasonableness

The Gradient Formula

For two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

This compares the vertical change to the horizontal change.

Example 1: Gradient from Two Points

Find the gradient of the line through A(1, 2) and B(5, 10).

Solution: \[m = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2\]

Example 2: Negative Gradient

Find the gradient of the line through C(-1, 5) and D(3, -3).

Solution: \[m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2\]

Example 3: Zero or Undefined

  • If \(y_1 = y_2\), then \(m = 0\) (horizontal line).
  • If \(x_1 = x_2\), the gradient is undefined (vertical line).

Practice Exercises

  1. Find the gradient of the line through:
    • E(2, 3) and F(6, 11)
    • G(-4, 1) and H(2, -5)
    • I(0, 4) and J(5, 4)
  2. Decide if the gradient is positive, negative, zero, or undefined:
    • K(3, -1) and L(3, 6)
    • M(-2, 4) and N(4, 4)
  3. A line has gradient \(-\frac{3}{2}\) and passes through (2, 1). Choose another point on the line.

Real-World Application

Gradient represents a rate of change. For example, a road that rises 1 meter for every 5 meters run has gradient \(\frac{1}{5}\).

Homework

Complete the gradient formula practice set.

  1. \(2\), \(-1\), \(0\)
  2. Undefined, zero
  3. Example: (0, 4) or (4, -2)

Previous Lesson: Lesson 5: Gradient Basics (Rise/Run)
Next Lesson: Lesson 7: Linear Equations and Intersections