We have looked at how to visualise a linear relationship on a number plane, and we learnt that we only actually need to identify two points on the number plane in order to sketch the line. We will now look at how to sketch a line directly from its equation, without needing to create a table of values first.
In Linear rules we learnt that all linear relationships can be expressed in the form: $y=mx+c$y=mx+c, where $m$m is equal to the change in the $y$yvalues for every increase in the $x$xvalue by $1$1, and $c$c is the value of $y$y when $x=0$x=0.
Lines drawn on the number plane, extend forever in both directions. If we ignore the special case of horizontal and vertical lines (which we will look at in another lesson), all other lines will either cross both the $x$xaxis and the $y$yaxis or they will pass through the origin, $\left(0,0\right)$(0,0).
Here are some examples:
We use the word intercept to refer to the point where the line crosses or intercepts with an axis.
The $y$yintercept is the point where the line crosses the $y$yaxis. The coordinates of the $y$yintercept will always have an $x$xcoordinate of zero.
Note: Every straight line must have at least one intercept but cannot have any more than two intercepts.
As mentioned previously, we only need to identify two points to sketch a a straight line, and the $x$x and $y$yintercepts are probably the most useful points to identify and plot. They are also two of the easier points to find as we are substituting in either the values $x=0$x=0 or $y=0$y=0, which simplifies the work needed to solve.
The $y$yintercept can be thought of as either the coordinate the point where the $y$yaxis is crossed, or simply the $y$yvalue at this point (as the $x$xvalue is by default $0$0).
Find the $y$yintercept for the straight line below:
The $y$yintercept is $6$−6, and the coordinates of the $y$yintercept are $\left(0,6\right)$(0,−6).
Consider the following graph.
State the $x$xvalue of the $x$xintercept.
State the $y$yvalue of the $y$yintercept.
Consider the linear equation $y=2x4$y=2x−4.
What are the coordinates of the $y$yintercept?
Give your answer in the form $\left(a,b\right)$(a,b).
What are the coordinates of the $x$xintercept?
Give your answer in the form $\left(a,b\right)$(a,b).
Now, sketch the line $y=2x4$y=2x−4:
The change in $y$yvalues for every increase in the $x$xvalue is called the gradient. The gradient is often thought of as the 'slope' of the line how steep it is.
The value of the gradient, $m$m, relates to the line as follows:
What is the gradient $m$m of the line $y=9x+3$y=9x+3?
Any straight line on the coordinate plane is defined entirely by its gradient and its $y$yintercept.
We can represent the equation of any straight line, except vertical lines, using what is known as the gradientintercept form of a straight line.
All linear relationships can be expressed in the form: $y=mx+c$y=mx+c.
We can use the applet below to see the effect of varying $m$m and $c$c on both the line and its equation.

In algebra, any number written immediately in front of a variable, is called a coefficient. For example, in the term $3x$3x, the coefficient of $x$x is $3$3. Any number by itself is known as a constant term.
In the gradientintercept form of a line, $y=mx+c$y=mx+c, the gradient, $m$m, is the coefficient of $x$x, and the $y$yintercept, $c$c, is a constant term.
Consider the line graph shown below:
The $x$xvalue at the $y$yintercept is $0$0.
What is the $y$yvalue at this point?
The equation of the line is $y=2x+3$y=−2x+3.
This is of the form $y=mx+c$y=mx+c.
Which pronumeral represents the $y$yintercept?
$y$y
$m$m
$x$x
$c$c
$y$y
$m$m
$x$x
$c$c
Consider the linear equation $y=2x+9$y=2x+9.
What are the values of the gradient $m$m and the $y$yintercept $c$c?
$m$m $=$= $\editable{}$
$c$c $=$= $\editable{}$
creates and displays number patterns; graphs and analyses linear relationships; and performs transformations on the Cartesian plane