0580

Linear Graphs

Coordinate Geometry and Graphs

What the number tells you

  • The gradient measures how steep a line is and which way it slopes
  • A gradient of 3 means the line rises 3 for every 1 step to the right, and a gradient of −4 means it falls 4 for every 1 step to the right
  • An uphill line, running from bottom left to top right, has a positive gradient; a downhill line has a negative one
  • Comparing steepness ignores the sign, so a gradient of −5 is steeper than one of 2
  • A horizontal line has gradient 0, and a vertical line has no gradient at all because there is no horizontal step to divide by

Measuring it

  • On a printed grid, pick two points that sit on the line and complete a right-angled triangle between them
A line drawn on a grid through (0, 2) and (1, 5), with a dashed right-angled triangle between those two points marking a rise of 5 − 2 = 3 against a run of 1 − 0 = 1, so the gradient is 3
Source: Gradient of a line by Save My Exams
  • The gradient is the vertical change divided by the horizontal change
  • Choose two points that sit exactly on grid intersections and are far apart, because a short triangle magnifies any reading error
  • Read the scales before counting squares: the two axes often use different scales, and one square is frequently not one unit
  • Put the minus sign back in yourself for a downhill line, since the counting gives only the size
A downhill line on a grid through (0, 1) and (1, −1), with a dashed triangle showing 2 down against 1 across, giving a gradient of −2
Source: Gradient of a line by Save My Exams
  • Away from a grid, the gradient between (x₁, y₁) and (x₂, y₂) is (y₂ − y₁) ÷ (x₂ − x₁)
    • Between (1, 3) and (5, 11): (11 − 3) ÷ (5 − 1) = 8 ÷ 4 = 2
  • Subtract the two coordinates in the same order top and bottom, or the sign comes out wrong
Exam tip

Read the sign off the picture before you calculate: a line going down from left to right has a negative gradient, whatever the arithmetic says. Then take the two differences in the same order top and bottom — reversing one of them is what turns a correct value negative.

Worked example

Finding a gradient from two points

A straight line passes through (1, 2) and (5, 10). Find its gradient.

Solution:

  • Gradient is the change up the side divided by the change along the bottom
  • Change in y: 10 − 2 = 8
  • Change in x: 5 − 1 = 4
  • Gradient = 8 ÷ 4 = 2
  • Take the two coordinates in the same order in both subtractions; reversing both gives (2 − 10) ÷ (1 − 5) = −8 ÷ −4 = 2, the same answer
  • Reversing only one of them gives −2, which is a line sloping the other way