Gradient of Linear Graphs

2026 Syllabus Objectives

By the end of these notes, you should be able to:

  1. Find the gradient of a straight line
  2. Find the gradient from a grid only
  3. Calculate the gradient of a straight line from the coordinates of two points on it

What is Gradient?

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Gradient is a measure of how steep a line is. It tells us how much a line goes up (or down) for every step it goes across.

Think of it like climbing stairs or a hill:

  • A steep hill has a large gradient
  • A gentle slope has a small gradient
  • A flat surface has zero gradient

In mathematics, we use the letter m to represent gradient.

Understanding Gradient

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The gradient of a line is calculated as:

Gradient = How much the line goes up ÷ How much the line goes across

Or, in mathematical language:

Gradient = Rise ÷ Run

What do "rise" and "run" mean?

  • Rise = the vertical change (how much the line goes up or down)
  • Run = the horizontal change (how much the line goes across)

Types of Gradient

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Lines can have different types of gradients:

1. Positive Gradient

  • The line slopes upwards from left to right
  • As you move to the right, the line goes up
  • Example: A line going uphill

2. Negative Gradient

  • The line slopes downwards from left to right
  • As you move to the right, the line goes down
  • Example: A line going downhill

3. Zero Gradient

  • The line is horizontal (flat)
  • There is no rise, only run
  • Gradient = 0

4. Undefined Gradient

  • The line is vertical (straight up and down)
  • There is rise but no run
  • We cannot calculate this gradient because we cannot divide by zero

Method 1: Finding Gradient from a Grid

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When you have a graph with a straight line drawn on it, you can find the gradient by counting squares on the grid.

Step-by-Step Method:

Step 1: Choose two clear points on the line where the line passes through grid corners (this makes counting easier)

Step 2: Draw a right-angled triangle using these two points:

  • Draw a horizontal line from one point
  • Draw a vertical line to meet the other point
  • The line itself forms the sloping side

Step 3: Count the squares:

  • Count how many squares up or down (this is the rise)
  • Count how many squares across (this is the run)

Step 4: Calculate the gradient:

  • Gradient = rise ÷ run

Step 5: Decide on the sign:

  • If the line slopes upwards (going left to right), the gradient is positive
  • If the line slopes downwards (going left to right), the gradient is negative

Example 1: Positive Gradient from a Grid

Imagine a line on a grid. You choose two points on the line:

  • Starting point at (1, 2)
  • Ending point at (4, 8)

Drawing a triangle between these points:

  • The line goes up 6 squares (rise = 6)
  • The line goes across 3 squares (run = 3)

Gradient = rise ÷ run = 6 ÷ 3 = 2

The gradient is positive because the line slopes upward.

Example 2: Negative Gradient from a Grid

Imagine another line. You choose two points:

  • Starting point at (2, 8)
  • Ending point at (6, 4)

Drawing a triangle:

  • The line goes down 4 squares (rise = -4, negative because it goes down)
  • The line goes across 4 squares (run = 4)

Gradient = rise ÷ run = -4 ÷ 4 = -1

The gradient is negative because the line slopes downward.

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