Straight line graphs

The graph of each of these equations is a straight line:

\(x = 3\)
\(y = 2\)
\(y = x\)
\(y = -2x\)
\(y = 3x - 1\)
\(x + y = 3\)
\(3x - 4y = 12\)
\(y - 2 = 3(x + 4) \)

If an equation can be rearranged into the form \(y = mx + c\), then its graph will be a straight line.

In the above:

\(x + y = 3\) can be rearranged as \(y = 3 - x\) (which can be re-written as \(y = −x + 3\)).

\(3x - 4y = 12 \) can be rearranged as \(y = \frac{3}{4}x - 3\).

\(y - 2 = 3(x + 4)\) can be rearranged as \(y = 3(x + 4) + 2 \) or \(y = 3x + 14\).

Vertical and horizontal lines

Vertical lines have equations of the form \(x = k\).

Horizontal lines have equations of the form \(y = c\).

Example

Draw the graph of \(x = 3\).

Mark some points on a grid which have an \(x\)-coordinate of 3, such as (3, 0), (3, 1), (3, -2).

The points lie on the vertical line \(x = 3\).

Plotting straight line graphs

A table of values can be used to plot straight line graphs.

Example

Draw the graph of \(y = 3x - 1\).

Create a table of values:

x	-1	0	1	2	3
y	\(\begin{array}{l} y = 3x - 1 \\ y = 3 \times - 1 - 1 \\ y = -3 - 1 \\ y = -4 \end{array}\)	\(\begin{array}{l} 3 \times 0 - 1 \\ = -1 \end{array}\)	2	5	8

x
-1
0
1
2
3

y
\(\begin{array}{l} y = 3x - 1 \\ y = 3 \times - 1 - 1 \\ y = -3 - 1 \\ y = -4 \end{array}\)
\(\begin{array}{l} 3 \times 0 - 1 \\ = -1 \end{array}\)
2
5
8

Plotting the coordinates and drawing a line through them gives:

This is the graph of \(y = 3x - 1\).

Sketching straight line graphs

If you recognise that the equation is that of a straight line graph, then it is not actually necessary to create a table of values.

Just two points are needed to draw a straight line graph, although it is a good idea to do a check with another point once you have drawn the graph.

Example

Draw the graph of \(y = 3x - 1\).

If you recognise this as a straight line then just choose two ‘easy’ values of \(x\), work out the corresponding values of \(y\) and plot those points.

When \(x = 0\), \(y = 3 \times 0 - 1 = −1\). Plot (0, −1).

When \(x = 2\), \(y = 3 \times 2 - 1 = 5\). Plot (2, 5).

Drawing the line through (0. -1) and (2, 5) gives the line above.

Example

Draw the graph of \(2x + 3y = 12\).

If you recognise this as a straight line then:

When \(x = 0\), then \(2 \times 0 + 3y = 12\) means \(3y = 12\), so \(y = 4\). Plot (0, 4).

When \(y = 0\), then \(2x + 3 \times 0 = 12\) means \(2x = 12\), so \(x = 6\). Plot (6, 0).

Draw the line through (0, 4) and (6, 0).

A straight line graph that plots the equation 2x + 3y = 12. The line is shown crossing number four on the Y axis and number six on the X axis.

Now check:

The drawn graph passes through (3, 2).

Does (3, 2) satisfy \(2x + 3y = 12?\)

\(2 \times 3 + 3 \times 2 \) does equal 12, so we can be confident that our line is correct.

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