This document covers straight lines and their properties including positive and negative gradient, the relationship between perpendicular lines, using graphs to determine speed and distance, and finding equations of lines. It provides examples of calculating gradient, determining if two lines are perpendicular, finding speed from a distance-time graph, and deriving equations of lines given points or being parallel/perpendicular to another line. The content builds understanding of key concepts involving straight lines through worked examples.

1. Unit 28
Straight Lines
Presentation 1 Positive and Negative Gradient
Presentation 2 Gradients of Perpendicular Lines
Presentation 3 Application of Graphs
Presentation 4 Equation of Straight Line

2. Unit 28
28.1 Positive and Negative
Gradient

3. A
B
Example
Find the gradient of the
line shown opposite.
Solution
-4 -3 -2 -1 0 1 2 3
5
4
3
2
1
-1
-2
-3
-4
-5
y
x
?
?
?
?
Vertical
change =
10
Horizontal
change =
6
x
y

4. A
B
Example
Find the gradient of the
line shown opposite.
Solution
x
y
-2 -1 0 1 2 3 4
5
4
3
2
1
A
B
(-2, 4)
(4, 1)
?
?
?
?
?
x
y

5. Unit 28
28.2 Gradient of Perpendicular
Lines

6. ?
If two lines are perpendicular to one another, then the product of
the two gradients is equal to -1.
So if is the gradient of one line , the other line has a
gradient of
Example
Show that the line segment joining the points A(3, 2) and B(5, 7) is
perpendicular to the line segment joining the points P(2, 5) and
Q(7, 3).
Solution
?
??
?
?
??
?
??

7. Unit 28
28.3 Application of Graphs

8. Distance-time
graph
Distance
Time
The gradient gives the velocity.
If the gradient is zero, the
object is not moving.
2000
1500
1000
500
0
50 100 150 200 250 300 350 400
Example
The graph shows the
distance travelled by a
girl on bicycle.
Find the speed she is
travelling on each stage
of the journey
Solution
?
?
?
?
?
?
?
?
?
A
B
C
D
E
Time (s)
Distance(m)

9. Solution
The distance is given by the area under the graph,
which can be split into 3 sections A, B and C
Velocity
Time
velocity-time
graph
The gradient gives the
acceleration.
If the gradient is zero, the object
is moving at a constant velocity.
The area under this graph is the
distance travelled.
8
7
6
5
4
3
2
1
2 4 6 8 10 12 14
Example
The graph shows how the
speed of a bird varies as it
flies between two trees. How
far apart are the two trees?A B C18m 36m 6m
?? ? ???? ???
Time (s)
Velocity(m/s)

10. Unit 28
28.4 Equation of a Straight Line

11. The equation a of a straight line is usually written in the form
Where m is the gradient and c is the intercept.
Example
(a)
(b) Equation of line AB:
As it passes through (3, 1)
and
10
8
6
4
2
-3 -2 -1 0 1 2 3 4 5 6
A
B
G
x
y
?
?
?
?
?
(-1, 9)
(3, 1)
?
??
???
? ??
O

12. 10
8
6
4
2
-3 -2 -1 0 1 2 3 4 5 6
A
B
G
x
y
(-1, 9)
(3, 1)
(c) Coordinates of G: so
(d) Equation of line through
O, the origin, perpendicular
to AB:
Equation:
i.e.
(e) Equation of line through O, parallel to AB
Equation:
i.e.
?
?
?
O
?
?
?
?
?
?
??