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
B
D
3
6
C
A
6
4
E
Gradient AB =
F
1
7
vertical = 6 = 1.5
4
horizontal
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 (X1,Y1)
and (X2,Y2).
Gradient =
Change in y
Change in x
=
Y2 - Y1
X2 - X1
(X2,Y2)
Y2 - Y1
(X1,Y1)
X2 - X1
Example:
Calculate the gradient of the line between the following
pairs of co-ordinates.
1. (1,2) and (5,18)
m =
Y2 - Y1
X2 - X1
= 18 - 2 = 16 =
4
5-1
4
Note: The gradient of a line is more usually given
the label (m).
2.
m =
3.
m =
(7,5) and (3,13)
Y2 - Y1
X2 - X1
= 13 - 5 =
3-7
8 =
-4
-2
(4, -2) and (-2, -5)
Y2 - Y1
X2 - X1
= -5 – (-2) = - 3 =
-6
-2-4
½
3.
(4, -2) and (-2, -5)
x
y
x (4, -2)
3
x
(-2, -5)
6