Gradient of an Interval
The ‘gradient’ of a line (or interval) is the measure of how steep
the line is.
Another word for gradient is ‘slope’
D
B
The interval AB is not as steep as
the interval CD
C
A
The gradient of AB < the gradient of CD
Gradient is defined as ‘the ratio
of the vertical rise to the
horizontal run between two
points on a line’
gradient  m 
vertical rise
horizontal run
Page 1
Gradient of an Interval
Positive and Negative gradient
As you move along a line from left to right:
B
Vertical
rise
A
If it slopes upwards, it has a
positive gradient.
Horizontal run
C
If it slopes downwards, it
has a negative gradient.
Vertical
rise
Horizontal run
D
Page 2
Gradient of an Interval
- Horizontal and Vertical Lines
y
A horizontal line has a
gradient of ‘zero’
6
5
In this case the
vertical rise is
zero
4
vertical rise
mAB 
horizontal run
0
mAB 
3
3
A
B
Horizontal run
3 units
2
mAB  0
x
1
y
2
3
4
5
6
A
A vertical line has an
‘infinite’ gradient
6
5
vertical
rise
3 units
4
3
In this case the
Horizontal run
is zero
2
mAB
B
x
1
2
vertical rise
horizontal run
3

0
mAB 
3
4
5
mAB  undefined
6
Page 3
Gradient of an Interval
- Positive, Negative or Neither
State whether each of the following lines has a ‘positive
gradient’, ‘negative gradient’ or ‘Neither’
y
y
1
y
2
3
x
x
Positive
x
Negative
y
y
4
y
5
x
Negative
Neither
6
x
Neither
x
Positive
Page 4
Gradient of an Interval
- Rise over Run
Calculate the gradient of each of the following lines.
A
1
A
2
6
3
A
2.5
8
B
5
B
B
8
3
mAB 
rise
run
mAB 
mAB 
6
3
mAB  
mAB  2
NB: Positive gradient
rise
run
mAB 
8
8
mAB 
mAB  1
NB: Negative gradient
mAB
rise
run
2.5
5
1

2
NB: Positive gradient
Page 5
Gradient of an Interval
- Rise over Run
Calculate the gradient of each of the following intervals.
1
y
2
B
y
8
8
7
7
6
6
6 units
5
4
3
3
A
5 units
5
4
2
A
B
7 units
2
3 units
1
1
x
1
mAB 
2
rise
run
3
4
5
6
mAB 
7
6
3
8
9
x
1
2
mAB 
3
rise
run
4
5
6
7
mAB  
8
9
5
7
mAB  2
Page 6
Gradient of an Interval
The gradient, m , of a
line passing through the
points A(x1,y1) and
B(x2,y2) is given by:
- The Gradient Formula
y
7
B( x2 , y2 )
6
Rise
5
4
mAB 
mAB 
vertical rise
horizontal run
y2  y1
x2  x1
y2  y1
3
Run
2
A( x1 , y1 )
C ( x2 , y1 )
x2  x1
1
y y
m 2 1
x2  x1
1
2
3
4
5
6
7
x
Page 7
Gradient of an Interval
The gradient, m , of a
line passing through
the points A(1 ,2) and
B(7 , 6) is given by:
- The Gradient Formula
y
7
B(7,6)
6
vertical rise
horizontal run
y2  y1

x2  x1
mAB 
mAB
62
mAB 
7 1
4
mAB 
6
2
mAB 
3
Rise
5
4
4units
3
Run
2
C (7, 2)
A(1,2)
6units
1
1
2
3
4
5
6
7
x
Page 8
Gradient of an Interval
- The Gradient Formula
y
The gradient, m , of a
line passing through
the points A(-2 , 3)
and B(3 ,-2) is given
by:
vertical rise
horizontal run
y2  y1

x2  x1
4
A
3
2
1
mAB 
mAB
3  (2)
2  3
5

5
mAB 
mAB
mAB  1
x
-4
-3
-2
-1
1
2
3
4
-1
-2
B
-3
-4
NB: The gradient will be negative
Page 9
Gradient of an Interval
- The Gradient Formula
When using the ‘gradient formula’ you don’t need to plot the two
points and draw the interval.
Just substitute straight into the ‘gradient formula’
Calculate the gradient of the
interval joining the points
A(7,3) and B(6,5)
mAB
y y
 2 1
x2  x1
mAB 
A( x1 , y1 )
B( x2 , y2 )
A( 7, 3)
B (6, 5)
y2  y1
x2  x1
53
67
2

1
mAB 
mAB
mAB  2
NB: The gradient is negative
Page 10
Gradient of an Interval
- Using the Gradient Formula
Try this one:
Calculate the gradient of the interval joining the points M(-3,6)
and N(4,-9)
mAB 
y2  y1
x2  x1
A( x1 , y1 )
B( x2 , y2 )
M (  3, 6) B (4,  9)
9  6
4  (3)
15

7
15

7
mMN 
mAB
mAB
NB: The gradient is negative
Page 11
Gradient of an Interval
- Using the Gradient Formula
Try this one:
The gradient of the interval joining the points C(-3,1) and D(-5, x)
is -2
Calculate the value of x
mCD 
y2  y1
x2  x1
x 1
5  (3)
x 1
2 
2
2 
C ( x1, y1 )
D( x2 , y2 )
C (  3, 1)
D(5, x)
mCD  2
Substituting values
Simplifying
4  x 1
Multiplying both sides by -2
5 x
Adding 1 to both sides
x5
Reversing sides
Therefore, point D has
co-ordinates (-5,5)
Page 12