Drawing linear graphs

Straight-line graphs

An equation such as $y=2x+1$ links two quantities. Every pair of values that makes the equation true is a point $(x,y)$: for example $x=3$ gives $y=7$, so $(3,7)$ is on the graph. The graph of the equation is the set of all such points.

For an equation of the form $y=mx+c$ the graph is always a straight line, which is why these are called linear equations. The reason is the steady step: every time $x$ increases by 1, $y$ changes by exactly $m$. A constant step in $y$ for each step in $x$ means a constant slope, and a constant slope is a straight line.

That steady step is also your best check. In a correct table for $y=2x+1$ the $y$-values go up by 2 each time; if one does not, it has been worked out wrongly.

Making a table of values

The question usually gives you the $x$-values, often in a table with some $y$-values already filled in. For each $x$:

1. Substitute it into the equation, putting negative values in brackets.
2. Work out $y$, doing the multiplication before the addition or subtraction.
3. Check that the $y$-values change by the same amount at each step.

In Core questions the equation is given in the form $y=mx+c$, such as $y=-2x+5$, unless a table of values is given. Take care with a negative $m$: for $y=-2x+5$ at $x=-1$, $-2\times (-1)=+2$, so $y=7$.

The completed graph for Worked example 1. Every plotted point lies exactly on the ruled line.

Drawing the line

Plot every point in your table with a small neat cross. Then:

• draw one straight line through all of them with a ruler, not a set of short pieces joined dot to dot;
• draw it across the whole range of $x$ the question asks for, from the first $x$-value to the last;
• label the line with its equation if there is more than one line on the grid.

Two points fix a line, but always plot at least three. If one point refuses to line up with the others, you have made an arithmetic slip: recalculate that $y$-value rather than bending the line.

Check the scales before you plot. If the $y$-axis goes up in 2s or 5s, a value like $y=7$ sits between grid lines.

Horizontal and vertical lines

Some linear equations have only one letter.

• $y=3$ is the set of points whose $y$-coordinate is 3, whatever $x$ is: $(-2,3)$, $(0,3)$, $(5,3)$. It is a horizontal line through 3 on the $y$-axis. It is $y=mx+c$ with $m=0$.
• $x=-2$ is the set of points whose $x$-coordinate is $-2$: a vertical line through $-2$ on the $x$-axis.

A common slip is to draw $y=3$ as a vertical line because it "crosses 3". Ask instead: which coordinate is fixed? If $y$ is fixed, the line runs across.

The horizontal line $y=3$ and the vertical line $x=-2$ cross at $(-2,3)$, the one point that is on both.

Using your graph

Once the line is drawn you can read from it.

• Find $y$ for a given $x$: go up (or down) from the $x$-value to the line, then across to the $y$-axis.
• Find $x$ for a given $y$: go across from the $y$-value to the line, then down to the $x$-axis.
• Where the line crosses the $x$-axis: read $x$ where $y=0$.

Draw faint guide lines to show your reading, because examiners give method marks for them. A reading from a graph is accepted within a small range, so read carefully using the small squares.

To test whether a point lies on a line without drawing, substitute its $x$-coordinate into the equation and see whether you get its $y$-coordinate.