Drawing Linear Graphs

What you'll learn

How to fill in a table of values from an equation such as $y = 3 x - 1$ .
How to plot coordinate points and draw a straight line through them.
How to read values from your graph, including decimals.
How to rewrite an equation like $x + y = 5$ before drawing it.

1. Coordinates: points on a grid

The horizontal number line is the $x$ -axis. The vertical number line is the $y$ -axis. Together, these are called the axes.

The coordinate axes meet at the origin, with the horizontal axis labelled x and the vertical axis labelled y.

The origin is the point where the axes cross, usually labelled O.

Definition

Coordinate point

A coordinate point is written as $(x, y)$ . The first number tells you how far to move along the $x$ -axis; the second tells you how far to move up or down on the $y$ -axis.

Example

Plotting coordinate points

To plot the point (2, -1), start at the origin.

Plotting coordinates means moving across first from the origin, then up or down.

Move 2 squares to the right because the $x$ -coordinate is 2.
Move 1 square down because the $y$ -coordinate is -1, then mark the point.
To plot (-1, 3), move 1 square left, then 3 squares up.

Common Mistake

Swapping the coordinates

Always go across first, then up or down. The point (3, -2) is not the same as (-2, 3).

Swapping the coordinates puts the point in a different position on the grid.

2. Completing a table of values

A table of values lists different $x$ -values and the matching $y$ -values.

Definition

Substitute

To substitute means to replace a letter, such as $x$ , with a number and then calculate the answer.

If the equation is $y = 3 x - 1$ , each $x$ -value gives one $y$ -value.

Example

Completing a table of values

Use the equation $y = 3 x - 1$ and the given $x$ -values: -2, -1, 0, 1, 2.
For $x = - 2$ , substitute -2 into the equation:
$y = 3 (- 2) - 1 = - 7$
For $x = 1$ , substitute 1 into the equation:
$y = 3 (1) - 1 = 2$
Complete the table:

x -2 -1 0 1 2
y -7 -4 -1 2 5

x	-2	-1	0	1	2
y	-7	-4	-1	2	5

Tip

Quick check at x = 0

In an equation like $y = m x + c$ , when $x = 0$ , the answer is the number on its own. For $y = 3 x - 1$ , the table should show -1 when $x = 0$ .

3. Drawing the straight-line graph

Definition

Linear graph

A linear graph is a straight-line graph. Equations like $y = 2 x + 1$ are linear because the $x$ is not squared or cubed.

The range of $x$ values means the smallest $x$ to the largest $x$ you have been asked to draw.

To draw the graph:

Make a table of values.
Plot the points.
Join the points with a ruler.
Only draw the line over the required range.

Example

Drawing a graph from a table

Draw the graph of $y = 2 x + 1$ for $x$ values from -3 to 3.

The points from the table for y = 2x + 1 lie on one straight line over the range -3 to 3.

Complete the table:

x -3 -2 -1 0 1 2 3
y -5 -3 -1 1 3 5 7
Plot the points, including (-3, -5), (0, 1), and (3, 7).
Use a ruler to draw one straight line through the points from $x = - 3$ to $x = 3$ .

x	-3	-2	-1	0	1	2	3
y	-5	-3	-1	1	3	5	7

Key Idea

Straight-line check

If your points do not lie in a straight line, recheck your table. One value may have been calculated or plotted incorrectly.

4. Handling negative and fractional equations

The coefficient is the number multiplying a variable. For example, in $- 3 x$ , the coefficient of $x$ is -3.

Some equations include negatives, like $y = 2 - 3 x$ . Some include fractions, like $y=12x+2y=\frac{1}{2}x+2$ . Work slowly and substitute carefully.

Example

Using a fractional coefficient

Use the equation $y=12x+2y=\frac{1}{2}x+2$ for $x$ values from -3 to 3.

Fractional y-values such as 0.5 and 1.5 are plotted halfway between grid lines.

For $x = - 3$ , half of -3 is -1.5:
$y=12(−3)+2=0.5y=\frac{1}{2}(-3)+2=0.5$
Complete the table:

x -3 -2 -1 0 1 2 3
y 0.5 1 1.5 2 2.5 3 3.5
Plot the points carefully. Some points are halfway between grid lines, so check the scale.

x	-3	-2	-1	0	1	2	3
y	0.5	1	1.5	2	2.5	3	3.5

Common Mistake

Forgetting the negative sign

In $2 - 3 x$ , if $x = - 2$ , then $- 3 x = 6$ , not -6. Multiplying two negatives gives a positive.

5. Reading values from a graph

The scale of an axis tells you what each grid square is worth. It might be 1 unit, 0.5 units, or 0.2 units.

To find a $y$ -value from a graph, start at the given $x$ -value. To find an $x$ -value, start at the given $y$ -value.

Example

Reading from a graph

Suppose you have drawn the graph of $y = 2 - 2 x$ and want to find $y$ when $x = 0.5$ .

To find y from a graph, start on the x-axis, move vertically to the line, then horizontally to the y-axis.

Find 0.5 on the $x$ -axis, move vertically to the line, then move horizontally to the $y$ -axis.
The value should be about 1, which you can check by substituting:
$y = 2 - 2 (0.5) = 1$
To find $x$ when $y = 3.5$ on the graph of $y=12x+2y=\frac{1}{2}x+2$ , start at 3.5 on the $y$ -axis, move across to the line, then down to the $x$ -axis.

To find x from a graph, start on the y-axis, move horizontally to the line, then vertically down to the x-axis.

The value should be about 3, which you can check from:
$3.5=12x+23.5=\frac{1}{2}x+2$

Tip

Use the graph like a route

For “find $y$ ”, start on the $x$ -axis. For “find $x$ ”, start on the $y$ -axis.

6. Equations like x + y = 5

Sometimes the equation is not written with $y$ on its own. Rearrange means change the equation into an equivalent form, usually with $y$ by itself.

Example

Rearranging before drawing

Start with the equation $x + y = 5$ .

After rearranging x + y = 5 to y = 5 - x, the graph is a decreasing straight line crossing the axes at 5.

Subtract $x$ from both sides to make $y$ the subject:
$y = 5 - x$
Complete a table for $x$ values from -1 to 5:

x -1 0 1 2 3 4 5
y 6 5 4 3 2 1 0
Plot the points and join them with a ruler.

x	-1	0	1	2	3	4	5
y	6	5	4	3	2	1	0

Common Mistake

Rearranging the wrong way

From $x + y = 5$ , the correct rearrangement is $y = 5 - x$ . It is not $y = x - 5$ .

Exam technique

In the exam

Check the required range of $x$ values before drawing the line.
Plot at least three points so you can spot a mistake more easily.
Use a ruler, label points carefully, and read the grid scale before estimating values.

Self review

Check yourself

Can you complete a table for $y = 4 - 3 x$ when $x = - 2, - 1, 0, 1, 2$ ?
Can you explain why a coordinate is plotted across first, then up or down?
If $x + y = 6$ , can you rewrite it with $y$ on its own?

Drawing Linear Graphs

What you'll learn

How to fill in a table of values from an equation such as $y = 3 x - 1$ .
How to plot coordinate points and draw a straight line through them.
How to read values from your graph, including decimals.
How to rewrite an equation like $x + y = 5$ before drawing it.

1. Coordinates: points on a grid

The horizontal number line is the $x$ -axis. The vertical number line is the $y$ -axis. Together, these are called the axes.

The coordinate axes meet at the origin, with the horizontal axis labelled x and the vertical axis labelled y.

The origin is the point where the axes cross, usually labelled O.

Definition

Coordinate point

A coordinate point is written as $(x, y)$ . The first number tells you how far to move along the $x$ -axis; the second tells you how far to move up or down on the $y$ -axis.

Example

Plotting coordinate points

To plot the point (2, -1), start at the origin.

Plotting coordinates means moving across first from the origin, then up or down.

Move 2 squares to the right because the $x$ -coordinate is 2.
Move 1 square down because the $y$ -coordinate is -1, then mark the point.
To plot (-1, 3), move 1 square left, then 3 squares up.

Common Mistake

Swapping the coordinates

Always go across first, then up or down. The point (3, -2) is not the same as (-2, 3).

Swapping the coordinates puts the point in a different position on the grid.

2. Completing a table of values

A table of values lists different $x$ -values and the matching $y$ -values.

Definition

Substitute

To substitute means to replace a letter, such as $x$ , with a number and then calculate the answer.

If the equation is $y = 3 x - 1$ , each $x$ -value gives one $y$ -value.

Example

Completing a table of values

Use the equation $y = 3 x - 1$ and the given $x$ -values: -2, -1, 0, 1, 2.
For $x = - 2$ , substitute -2 into the equation:
$y = 3 (- 2) - 1 = - 7$
For $x = 1$ , substitute 1 into the equation:
$y = 3 (1) - 1 = 2$
Complete the table:

x -2 -1 0 1 2
y -7 -4 -1 2 5

x	-2	-1	0	1	2
y	-7	-4	-1	2	5

Tip

Quick check at x = 0

In an equation like $y = m x + c$ , when $x = 0$ , the answer is the number on its own. For $y = 3 x - 1$ , the table should show -1 when $x = 0$ .

3. Drawing the straight-line graph

Definition

Linear graph

A linear graph is a straight-line graph. Equations like $y = 2 x + 1$ are linear because the $x$ is not squared or cubed.

The range of $x$ values means the smallest $x$ to the largest $x$ you have been asked to draw.

To draw the graph:

Make a table of values.
Plot the points.
Join the points with a ruler.
Only draw the line over the required range.

Example

Drawing a graph from a table

Draw the graph of $y = 2 x + 1$ for $x$ values from -3 to 3.

The points from the table for y = 2x + 1 lie on one straight line over the range -3 to 3.

Complete the table:

x -3 -2 -1 0 1 2 3
y -5 -3 -1 1 3 5 7
Plot the points, including (-3, -5), (0, 1), and (3, 7).
Use a ruler to draw one straight line through the points from $x = - 3$ to $x = 3$ .

x	-3	-2	-1	0	1	2	3
y	-5	-3	-1	1	3	5	7

Key Idea

Straight-line check

If your points do not lie in a straight line, recheck your table. One value may have been calculated or plotted incorrectly.

4. Handling negative and fractional equations

The coefficient is the number multiplying a variable. For example, in $- 3 x$ , the coefficient of $x$ is -3.

Some equations include negatives, like $y = 2 - 3 x$ . Some include fractions, like $y=12x+2y=\frac{1}{2}x+2$ . Work slowly and substitute carefully.

Example

Using a fractional coefficient

Use the equation $y=12x+2y=\frac{1}{2}x+2$ for $x$ values from -3 to 3.

Fractional y-values such as 0.5 and 1.5 are plotted halfway between grid lines.

For $x = - 3$ , half of -3 is -1.5:
$y=12(−3)+2=0.5y=\frac{1}{2}(-3)+2=0.5$
Complete the table:

x -3 -2 -1 0 1 2 3
y 0.5 1 1.5 2 2.5 3 3.5
Plot the points carefully. Some points are halfway between grid lines, so check the scale.

x	-3	-2	-1	0	1	2	3
y	0.5	1	1.5	2	2.5	3	3.5

Common Mistake

Forgetting the negative sign

In $2 - 3 x$ , if $x = - 2$ , then $- 3 x = 6$ , not -6. Multiplying two negatives gives a positive.

5. Reading values from a graph

The scale of an axis tells you what each grid square is worth. It might be 1 unit, 0.5 units, or 0.2 units.

To find a $y$ -value from a graph, start at the given $x$ -value. To find an $x$ -value, start at the given $y$ -value.

Example

Reading from a graph

Suppose you have drawn the graph of $y = 2 - 2 x$ and want to find $y$ when $x = 0.5$ .

To find y from a graph, start on the x-axis, move vertically to the line, then horizontally to the y-axis.

Find 0.5 on the $x$ -axis, move vertically to the line, then move horizontally to the $y$ -axis.
The value should be about 1, which you can check by substituting:
$y = 2 - 2 (0.5) = 1$
To find $x$ when $y = 3.5$ on the graph of $y=12x+2y=\frac{1}{2}x+2$ , start at 3.5 on the $y$ -axis, move across to the line, then down to the $x$ -axis.

To find x from a graph, start on the y-axis, move horizontally to the line, then vertically down to the x-axis.

The value should be about 3, which you can check from:
$3.5=12x+23.5=\frac{1}{2}x+2$

Tip

Use the graph like a route

For “find $y$ ”, start on the $x$ -axis. For “find $x$ ”, start on the $y$ -axis.

6. Equations like x + y = 5

Sometimes the equation is not written with $y$ on its own. Rearrange means change the equation into an equivalent form, usually with $y$ by itself.

Example

Rearranging before drawing

Start with the equation $x + y = 5$ .

After rearranging x + y = 5 to y = 5 - x, the graph is a decreasing straight line crossing the axes at 5.

Subtract $x$ from both sides to make $y$ the subject:
$y = 5 - x$
Complete a table for $x$ values from -1 to 5:

x -1 0 1 2 3 4 5
y 6 5 4 3 2 1 0
Plot the points and join them with a ruler.

x	-1	0	1	2	3	4	5
y	6	5	4	3	2	1	0

Common Mistake

Rearranging the wrong way

From $x + y = 5$ , the correct rearrangement is $y = 5 - x$ . It is not $y = x - 5$ .

Exam technique

In the exam

Check the required range of $x$ values before drawing the line.
Plot at least three points so you can spot a mistake more easily.
Use a ruler, label points carefully, and read the grid scale before estimating values.

Self review

Check yourself

Can you complete a table for $y = 4 - 3 x$ when $x = - 2, - 1, 0, 1, 2$ ?
Can you explain why a coordinate is plotted across first, then up or down?
If $x + y = 6$ , can you rewrite it with $y$ on its own?

Drawing Linear Graphs

What you'll learn

1. Coordinates: points on a grid

2. Completing a table of values

3. Drawing the straight-line graph

4. Handling negative and fractional equations

5. Reading values from a graph

6. Equations like x + y = 5

Recap questions

You've reached the end

Drawing Linear Graphs

What you'll learn

1. Coordinates: points on a grid

2. Completing a table of values

3. Drawing the straight-line graph

4. Handling negative and fractional equations

5. Reading values from a graph

6. Equations like x + y = 5

Recap questions

You've reached the end

Drawing Linear Graphs

What you'll learn

1. Coordinates: points on a grid

2. Completing a table of values

3. Drawing the straight-line graph

4. Handling negative and fractional equations

5. Reading values from a graph

6. Equations like x + y = 5

Recap questions

You've reached the end

Drawing Linear Graphs Revision Guide

Drawing Linear Graphs

What you'll learn

1. Coordinates: points on a grid

2. Completing a table of values

3. Drawing the straight-line graph

4. Handling negative and fractional equations

5. Reading values from a graph

6. Equations like x + y = 5

Recap questions

You've reached the end

Drawing Linear Graphs Revision Guide