GCSE Mathematics (1-9) 2015
How To Do it
A10Theory
Algebra - Gradients and intercepts - 10
identify and interpret gradients and intercepts of linear functions graphically and
algebraically
See A9 also
1
Gradients
The slope, or gradient, of a line, is how
steep it is. We measure the slope as
5
distance up
distance along
For the line shown, if we go along 3, we
go up 6, so the gradient = 6/3 = 2
For a line that slopes down, the distance
up is negative, so the gradient is
negative.
-5
0
5
-5
If the line is of the form
y = mx + c
the gradient is m. For example for the line
y = 4x -7 the gradient is 4. For
y = 3 - 2x the gradient is -2 : it slopes down.
2
Intercepts
The intercepts are the distances along the
x and y axis where the lines cuts the axis,
to the origin.
The line shown has y intercept 5, and x
intercept -4
If the line has the form
5
x intercept
0
-5
y = mx+c
then the y intercept is c. For example
y = 2x + 5 has y intercept 5
-5
y intercept
5
GCSE Mathematics (1-9) 2015
How To Do it
A10Theory
The x intercept is the x value when y = 0. So if
y = mx +c
this is when
0 = mx+c
so x = -c/m
For example if
y = 2x+5
the x intercept would be
x = -5/2
3
Linear functions : intercept
Functions of of the form y = mx + c, where m and c are constants, are called linear
functions, because their graphs are straight lines.
If we draw graphs of y=2x+c, with
different values of c, we get this:
y=2x+2
5
So graphs of
y=2x
y=2x-3
y = 2x + c
cross the y axis at y=c. This is called
intercept.
-5
y=2x+4
0
-5
5
the
GCSE Mathematics (1-9) 2015
4
How To Do it
A10Theory
Linear functions - slope
The slope of a line is how much it rises, when you move along one step:
slope=-2
slope=3
5
slope=2
slope=1
0
-5
5
-5
If we plot several lines like
y=mx for different values of m, we
they have slope m.
y=2x
slope=2
y=x
slope=1
5
y=3x
slope=3
So a function of the form
-5
0
y = mx + c
has slope m and intercept c
-5
intercept c
slope m
5
Parallel and perpendicular lines
Parallel lines have the same slope. So functions in teh form
y == mx + c
have are parallel if their values for m are the same.
For example
y = 3x +2
is parallel to y = 3x -7
5
find
GCSE Mathematics (1-9) 2015
How To Do it
A10Theory
This might need some re-arranging. For example
is
2y -12 = 4x ..{line 1}
parallel to
y=2x-4 .. {line 2}
add 12 to both sides of line 1
2y= 4x+12
divide both sides by 2
y = 2y +6
this has the same m value as line 2, so they are parallel.
Two lines are perpendicular if the product of their slopes is -1. So for example
y = 2x -3
and
y = 4 -x/2
are perpendicular, since 2 X ( -1/2) = -1
6
Line through a given point
For example
(y-3) = 4 (x-2)
goes through the point (2,3). If x=2, the RHS is zero, and so is the LHS, so (2,3) is on teh
line. The line has slope 4.
In general y-y 1 = m(x-x 1 ) goes through (x 1 , y 1 ) and has slope m.
7
Line through two points
If a line goes through (x1,y1) and (x2,y2), then it moves along (x2-x1) as it goes up from
y1 up to y2.
So its slope is
y1 − y2
x1 −x2
and the function is
y− y 1=
y 1− y 2
( x−x 1)
x 1−x 2
GCSE Mathematics (1-9) 2015
How To Do it
A10Theory