The Linear Function
The Linear Equation
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Linear equations have both variables of power one and no
variables multiplied together.
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In general straight line functions can be represented in a
number of forms. Typically, the equation of a straight line can
be expressed as
y = mx + c ,
The Learning Centre
where c is the y -intercept and m is the slope or the gradient
of the line.
Examples:
y
y
y
y
y
y
y
= 7x
= −3x
= 0.5x
= x5
= 7x + 3
= −3x + 7
= 0.5x − 1
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Lines with the same slope are parallel lines.
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Lines with a point in common are called concurrent or
intersecting lines.
Answers:
:
:
:
:
:
:
:
m=
m=
m=
m=
m=
m=
m=
,
,
,
,
,
,
,
c
c
c
c
c
c
c
=
=
=
=
=
=
=
.
.
.
.
.
.
.
y
y
y
y
y
y
y
= 7x
= −3x
= 0.5x
= x5
= 7x + 3
= −3x + 7
= 0.5x − 1
:
:
:
:
:
:
:
m = 7,
m = −3 ,
m = 0.5 ,
m = 15 ,
m = 7,
m = −3 ,
m = 0.5 ,
c
c
c
c
c
c
c
= 0.
= 0.
= 0.
= 0.
= 3.
= 7.
= −1 .
Summary
Let us summarise what we have learnt so far about linear
equations.
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If m is positive, the gradient is positive and the line rises as
we move from left to right.
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If m is negative, the gradient is negative and the line falls as
we move from left to right.
Example
Find the slope and y -intercept of the following equation.
y = −3x + 2 .
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The greater the size of m the steeper the line.
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Parallel lines have the same gradient.
Firstly, see that it is a linear equation. We should next check that
it is in the form y = mx + c .
In this case it is in the correct form, therefore we read off the
values for m and c.
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The point where the line cuts the y -axis is called the
y -intercept and is represented by the letter c.
Gradient = −3 .
Lines parallel to the y -axis have infinite slope while lines
parallel to the x-axis have zero slope.
y -intercept = 2 .
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This information now gives us a very powerful tool for estimating
what a linear graph will look like based on its equation.
Exercise
Find the slopes and intercepts of the lines below and sketch their
graphs.
1. y = 3x + 2
2. y = 3 − x
3. x = 10 − 2y
4. 3x − 4y − 5 = 0
Solutions:
1. y intercept is 2, slope is 3.
2. y intercept is 3, slope is −1.
3. Rearranging gives y = − 12 x + 5 in slope-intercept form.Thus,
the y intercept is 5, slope is − 12 .
4. Rearranging gives y = − 34 x − 54 in slope-intercept form.Thus,
the y intercept is − 54 , slope is − 34 .
Slope-point equation
Solutions
1.
If the slope, m, of a line and one point, say (x1 , y1 ), on the line are
known, then the equation of the line may be found as follows:
y = y1 + m(x − x1 ) .
y
= −1 + 1(x − 1)
y
= −2 + x .
y
= y1 + m(x − x1 )
1
= 0 − (x − 2)
2
1
= 1− x.
2
y
1. passing through (1, −1) with slope 1
2. passing through (2, 0) with slope − 12
y
3.
Note: You should always check your answer by substituting x1 for
x in the equation.
Horizontal lines
= y1 + m(x − x1 )
2.
Find the equations of the straight lines
3. passing through (−3, 1) with slope 2.
y
y
= y1 + m(x − x1 )
y
= 1 + 2(x − (−3))
y
= 7 + 2x .
Vertical lines
The gradient of a line parallel to the x-axis is zero.
The gradient of a line parallel to the y -axis is infinity.
Notice that the value for y in the figure above is 2 no matter what
x value we choose. We say that the equation of this line is y = 2 .
Notice that the value for x in the figure above is 1 no matter what
y value we choose. We say that the equation of this line is x = 1 .
Any line parallel to the x-axis will have an equation in the form
y = a.
Any line parallel to the y -axis will have an equation in the form
x = b.