Mathematics SKE: STRAND G
UNIT G2 Straight Lines: Introduction
G2 Straight Lines
Introduction
Learning objectives
This is the second unit in this strand, focusing on the straight line. After completing Unit G2 you
should
•
*•
•
*•
be able to identify and use the gradient of a straight line
be able to calculate and identify gradients of perpendicular lines
be confident in applying the concept of straight line graphs in a variety of situations, including
distance, time, velocity and acceleration
be able to develop and use the general equation of a straight line.
Introduction
The introduction to this topic is given in Unit G1, Coordinates. Here we focus on straight lines from
Geometry and use algebraic notation to find and apply the general equation of a straight line.
y
y
Key points and principles
• The gradient of a line can be positive or negative.
negative
gradient
positive
gradient
O
O
x
x
• Parallel lines have the same gradient.
Distance
• 'Perpendicular lines' means that the product of their
1
gradients equals −1 (i.e. m and − ).
m
• The gradient in a distance-time graph is the velocity.
O
Note: if the gradient is zero, the object is not moving.
Time
Velocity
• The gradient in a velocity-time graph is the acceleration.
Note: if the gradient is zero, the object is moving with
constant velocity; if the gradient is negative,
it is decelerating.
O
Time
y
y = mx + c
gradient, m
• The area under a velocity-time graph is the distance travelled.
• The general equation of a straight line is of the form
y = mx + c
where m is its gradient and c the y-axis intercept.
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1
intercept, c
c
O
x
Mathematics SKE: STRAND G
UNIT G2 Straight Lines: Introduction
G2 Straight Lines
Introduction
Facts to remember
•
y
The gradient of a straight line
vertical change
(y2 - y1)
y2 − y1
x2 − x1
=
(x2, y2)
y2
vertical change
=
horizontal change
(x1, y1)
y1
O
x1
x2
horizontal change
(x2 - x1)
x
•
Parallel lines have equal gradients.
•
The product of the gradients of perpendicular lines is −1; if m is the gradient of one of the
1
lines ⎛ − ⎞ is the gradient of the other line.
⎝ m⎠
•
The gradient of a distance-time graph is velocity.
•
The gradient of a velocity-time graph is acceleration.
•
The area under a velocity-time graph is the distance covered.
•
The general equation of a straight line is
y
y = mx + c
y = mx + c
where m is the gradient and c is the y-axis intercept.
gradient
m
intercept
c
c
x
O
Glossary of terms
y
Gradient of a straight line describes how steep the line is
and is defined as
m=
=
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(x2, y2)
y2
vertical change
(y2 - y1)
vertical change
horizontal change
(x1, y1)
y1
y2 − y1
x2 − x1
O
2
x1
x2
horizontal change
(x2 - x1)
x
Mathematics SKE: STRAND G
UNIT G2 Straight Lines: Introduction
G2 Straight Lines
Parallel lines
Introduction
lines with the same gradient are parallel to
one another.
y
x
O
Perpendicular lines
y
the product of their gradients is −1 ;
for example, see diagram.
slope 2
x
O
slope −
Distance-time graph
here the gradient represents the velocity;
the velocity is zero when the line has
zero gradient (for example, A to B in
the diagram opposite).
Distance
A
O
Velocity-time graph
here the gradient represents the
acceleration; the area under the
graph is the distance travelled.
1
2
B
Time
Velocity
Distance
travelled
General equation of a straight line is
Time
y
y = mx + c
y = mx + c
when m is the gradient and c is
the y-axis intercept.
gradient
m
intercept
c
c
O
© CIMT, Plymouth University
3
x