The gradient of a line The gradient of a line is a specific term for the steepness of a straight line. We all have an in-built sense of steepness and can order the steepness of lines. However, the gradient gives a numerical value to this general understanding.

To calculate the gradient of a line we count the vertical distance that line increases and horizontal distance that the line and then use the following calculation. gradient = vertical change horizontal change

4 6 A B 6 3 C D 7 1 E F Gradient AB = vertical horizontal = 6 4 =1.5

A B C

Note: A negative gradient means that the line is travelling downhill or a decline.

Calculating the gradient from co-ordinates It is possible to calculate the gradient of a line just by knowing two co-ordinates that the line passes through. This can be achieved in two ways: 1.Draw the co-ordinates on a grid and use the previous method. Gradient = Change in y Change in x Vertical horizontal =

2. Using a formula that has been specifically generated for the calculation. Let a line pass through two co-ordinates (X 1,Y 1 ) and (X 2,Y 2 ). Gradient = Change in y Change in x = Y 2 - Y 1 X 2 - X 1

(X 1,Y 1 ) (X 2,Y 2 ) Y 2 - Y 1 X 2 - X 1

Example: Calculate the gradient of the line between the following pairs of co-ordinates. 1. (1,2) and (5,18) Note: The gradient of a line is more usually given the label (m). Y 2 - Y 1 X 2 - X 1 = m = = 16 4 =4

2.(7,5) and (3,13) Y 2 - Y 1 X 2 - X 1 = m = = 8 -4 =-2 3.(4, -2) and (-2, -5) Y 2 - Y 1 X 2 - X 1 = m -5 – (-2) = = = ½

3.(4, -2) and (-2, -5) x (4, -2) x (-2, -5) x y 3 6