Why is "Reading coordinates the wrong way round, e.g. plotting $(3, 2)$ as 2 across and 3 up." a common mistake in Linear functions and Graphs?

Why it happens: Without a memory hook, it is easy to forget which number comes first. How to avoid it: Use 'along the corridor, then up the stairs' — the x-number always comes first.

Why is "Moving the wrong way for a negative coordinate, e.g. plotting $(-2, 3)$ to the right." a common mistake in Linear functions and Graphs?

Why it happens: Students count squares but forget the sign tells them the direction. How to avoid it: A negative x means move left; a negative y means move down.

Why is "Applying a function rule in the wrong order, e.g. $y = 2x + 1$ with $x = 4$ giving $2 \times (4 + 1) = 10$." a common mistake in Linear functions and Graphs?

Why it happens: The order of operations is overlooked when working through a rule. How to avoid it: Follow BIDMAS: multiply first, then add. 2 × 4 + 1 = 9.

Why is "Joining graph points with a wobbly freehand line instead of a ruler." a common mistake in Linear functions and Graphs?

Why it happens: It feels quicker to sketch the line by hand. How to avoid it: A linear function makes a perfectly straight line — always use a ruler to join the points.

Why is "Drawing the line through a point that does not line up with the others." a common mistake in Linear functions and Graphs?

Why it happens: Every plotted point is assumed to be correct. How to avoid it: If one point is off the line, re-check it in the table — a linear graph has all points in a straight row.

AlgebraCambridge Lower Secondary Mathematics (Grade 6)

Linear functions and Graphs

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Linear functions and graphs — Cambridge Lower Secondary Maths, Grade 6

A graph turns numbers into a picture you can read at a glance. This guide walks you through coordinates in all four quadrants, plotting points carefully, drawing and reading straight-line graphs, and the idea of a function as a number machine — one short step at a time.

What you’ll learn

Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.

Stage 7 — Read and plot coordinates in all four quadrants of a grid.
Stage 7 — Understand a function as a rule that maps each input to exactly one output.
Stage 7 — Generate a table of values from a function rule and plot the points.
Stage 7 — Draw and read straight-line graphs.

Coordinates and the four quadrants

Coordinates pinpoint a place on a grid using two numbers — across, then up.

Coordinates are a pair of numbers that name an exact point on a grid, written as $(x, y)$ .

The grid is built from two number lines called axes. The horizontal one is the $x$ -axis and the vertical one is the $y$ -axis. They cross at the origin, the point $(0, 0)$ .

The golden rule for reading coordinates is across before up — the $x$ -number first, the $y$ -number second. A useful memory hook: "along the corridor, then up the stairs".

Always read across the x-axis first, then up the y-axis.

The two axes cut the grid into four quadrants. In some of them, $x$ or $y$ (or both) are negative. So a point can be $(3, 2)$ , but it can also be $(-2, -3)$ or $(-4, 1)$ . The numbers tell you exactly where to go: a negative $x$ means go left, a negative $y$ means go down.

Coordinates are written $(x, y)$ — across, then up.
The axes are the $x$ -axis and $y$ -axis; they cross at the origin.
The origin is the point $(0, 0)$ .
Four quadrants allow negative $x$ and $y$ values.

Plotting points accurately

Start at the origin, count across for $x$ , then count up or down for $y$ .

Plotting a point means marking it in the right place on the grid. Here is a reliable method that always works:

Start at the origin $(0, 0)$ .
Move along the $x$ -axis by the first number — right if positive, left if negative.
From there, move parallel to the $y$ -axis by the second number — up if positive, down if negative.
Mark a small, neat dot or cross.

Across 4, then up 3 — the point lands at (4, 3).

To read a point that is already plotted, do the same in reverse: see how far it is along the $x$ -axis, then how far up the $y$ -axis.

Accuracy matters. A point plotted one square out will pull your whole graph out of shape. Take your time, count carefully, and keep your dots small so they sit exactly on the grid line crossings.

Always start counting from the origin.
Move across for $x$ first, then up or down for $y$ .
Negative numbers mean go left or down.
Keep marks small and neat for accuracy.

Functions — number machines

A function takes an input and gives back exactly one output, following a rule.

A function is a rule that turns one number into another. The best picture is a number machine: you feed an input in, the machine follows its rule, and an output comes out.

Input 4, apply "×2 then +1", output 9.

The machine above follows the rule "multiply by 2, then add 1". Feed in 4 and you get $4 \times 2 + 1 = 9$ .

We often write a function as an equation. If the input is $x$ and the output is $y$ , the machine above is $y = 2x + 1$ .

The one strict rule of a function: each input gives exactly one output. You can never put 4 in and get two different answers out.

A function is also called a mapping, because it maps (sends) each input to its matching output. Functions, mappings and number machines are three names for the same idea.

A function turns an input into one output, by a rule.
Picture a number machine: input → rule → output.
$y = 2x + 1$ means output $= 2 \times$ input $+ 1$ .
Each input gives exactly one output.

From function to table of values

Feed several inputs through the rule to build a neat table of $(x, y)$ pairs.

Before you can draw a graph, you need some points. A table of values is the tidy way to collect them.

Pick a few input values for $x$ , run each through the function rule, and record the output $y$ . Each column of the table is then a coordinate pair $(x, y)$ .

Take the function $y = 2x + 1$ :

Each column is a coordinate pair: (0,1), (1,3), (2,5), (3,7).

To fill the table:

When $x = 0$ : $y = 2 \times 0 + 1 = 1$ .
When $x = 1$ : $y = 2 \times 1 + 1 = 3$ .
When $x = 2$ : $y = 2 \times 2 + 1 = 5$ .
When $x = 3$ : $y = 2 \times 3 + 1 = 7$ .

Notice how $y$ goes up by 2 each time $x$ goes up by 1 — that steady jump is a sign the points will form a straight line. A neat table makes plotting quick and almost mistake-proof.

A table of values lists inputs and their outputs.
Run each $x$ through the rule to find $y$ .
Each column is a coordinate pair $(x, y)$ .
A steady jump in $y$ hints at a straight line.

Drawing and reading straight-line graphs

Plot the points from your table, join them with a ruler, and read values off the line.

Now turn the table into a picture. Plot each coordinate pair, then join the points with a straight line using a ruler. A function like $y = 2x + 1$ is linear — its graph is always a straight line.

The plotted points line up perfectly — a linear function.

A neatly drawn graph is a tool you can read in two directions:

From $x$ to $y$ : pick a value on the $x$ -axis, go straight up to the line, then across to the $y$ -axis.
From $y$ to $x$ : start on the $y$ -axis, go across to the line, then down to the $x$ -axis.

This is why graphs are so useful — you can find an answer without doing the calculation. If your points do not line up, that is a clear signal to check your table: a straight-line function must give points in a straight row.

Plot every point from the table.
Join them with a ruler — a linear function gives a straight line.
Read from $x$ to $y$ : up to the line, then across.
Read from $y$ to $x$ : across to the line, then down.

Where you'll use this next

Coordinates and graphs are the language of much of the maths still to come.

Getting confident with coordinates and graphs now helps across many topics:

Algebra and graphs meet in equations like $y = mx + c$ , where the numbers control the line's steepness and starting point.
Sequences can be plotted as points to reveal their pattern visually.
Statistics uses the same grid skills for scatter graphs and line graphs of real data.
Geometry uses coordinates to describe shapes, transformations and distances.

In everyday life, graphs are everywhere — temperature charts, sports performance trackers, currency converters and the graphs you see in the news all rely on exactly these skills.

If graph work feels hard later, it is usually a coordinate-reading slip underneath. Come back to this guide whenever you need a refresher — there is no prize for rushing.

Algebra and graphs meet in equations like $y = mx + c$ .
Sequences can be plotted to show their pattern.
Statistics uses grids for scatter and line graphs.
Geometry uses coordinates to describe shapes.

Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 7). Last reviewed 2026-05-19.

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Step-by-step worked examples — Linear functions and Graphs

Step-by-step solutions for linear functions and graphs, written exactly the way a tutor would explain them at the board.

1Reading coordinates from a grid
Getting started• coordinates, grid
Question
A point is 5 squares right of the origin and 2 squares down. Write its coordinates.
Step-by-step solution
1. Step 1
  The $x$ -coordinate comes first. The point is 5 squares right, so $x = 5$ .
2. Step 2
  The $y$ -coordinate comes second. The point is 2 squares down, so $y = -2$ .
3. Step 3
  Write the pair with $x$ first.
  $(5, -2)$
Answer
$(5, -2)$
2Using a function rule
Getting started• function, rule
Question
A function has the rule $y = 3x - 2$ . Find the output when the input is $x = 4$ .
Step-by-step solution
1. Step 1
  Substitute $x = 4$ into the rule.
  $y = 3 \times 4 - 2$
2. Step 2
  Multiply before you subtract.
  $y = 12 - 2$
3. Step 3
  Work out the final value.
  $y = 10$
Answer
$y = 10$
3Making a table of values
Building confidence• table of values, function
Question
Complete a table of values for $y = x + 3$ using the inputs $x = 0, 1, 2, 3$ .
Step-by-step solution
1. Step 1
  When $x = 0$ , apply the rule.
  $y = 0 + 3 = 3$
2. Step 2
  When $x = 1$ , apply the rule.
  $y = 1 + 3 = 4$
3. Step 3
  When $x = 2$ and $x = 3$ , do the same.
  $y = 5 \text{ and } y = 6$
4. Step 4
  The coordinate pairs are $(0,3)$ , $(1,4)$ , $(2,5)$ and $(3,6)$ .
Answer
$y$ -values: 3, 4, 5, 6
4Plotting points in four quadrants
Building confidence• plotting, quadrants
Question
Describe how to plot the point $(-3, 4)$ on a coordinate grid.
Step-by-step solution
1. Step 1
  Start at the origin $(0, 0)$ .
2. Step 2
  The $x$ -coordinate is $-3$ , so move 3 squares to the left.
3. Step 3
  The $y$ -coordinate is $4$ , so move 4 squares up from there.
4. Step 4
  Mark a small neat dot — the point sits in the top-left quadrant.
Answer
Left 3, up 4 — in the top-left quadrant.
5Reading a value from a graph
Stretch• straight-line graph, reading graphs
Question
The straight line $y = 2x + 1$ has been drawn. Use the graph to find the value of $y$ when $x = 5$ , then check it with the rule.
Step-by-step solution
1. Step 1
  On the graph, find 5 on the $x$ -axis and go straight up to the line.
2. Step 2
  From that point on the line, go straight across to the $y$ -axis to read the value.
3. Step 3
  Check using the rule: substitute $x = 5$ .
  $y = 2 \times 5 + 1 = 11$
4. Step 4
  The graph reading and the calculation agree, so $y = 11$ .
Answer
$y = 11$

Key Definitions and Keywords — Linear functions and Graphs

The important words for linear functions and graphs and what they mean — learn these so you can explain your thinking clearly.

Coordinates
Key word
A pair of numbers $(x, y)$ that gives the exact position of a point on a grid.
Example
$(3, 2)$ means 3 across and 2 up.
x-axis
The horizontal number line on a coordinate grid.
y-axis
The vertical number line on a coordinate grid.
Origin
Key word
The point $(0, 0)$ where the $x$ -axis and $y$ -axis cross.
Quadrant
Key word
One of the four regions a coordinate grid is divided into by the two axes.
Function
Key word
A rule that takes each input and gives exactly one output.
Example
$y = 2x + 1$ is a function.
Input
The number that goes into a function before the rule is applied.
Output
The number that comes out of a function after the rule is applied.
Mapping
Another word for a function — it maps each input to its matching output.
Table of values
Key word
A table listing inputs and the outputs found from a function rule.
Linear function
A function whose graph is a straight line, such as $y = 2x + 1$ .
Plot
To mark a point on a grid in the position given by its coordinates.

Common Mistakes and Misconceptions — Linear functions and Graphs

The slip-ups students most often make with linear functions and graphs — and simple ways to avoid them.

✕Reading coordinates the wrong way round, e.g. plotting $(3, 2)$ as 2 across and 3 up.
Why it happens
Without a memory hook, it is easy to forget which number comes first.
How to avoid it
Use 'along the corridor, then up the stairs' — the $x$ -number always comes first.
✕Moving the wrong way for a negative coordinate, e.g. plotting $(-2, 3)$ to the right.
Why it happens
Students count squares but forget the sign tells them the direction.
How to avoid it
A negative $x$ means move left; a negative $y$ means move down.
✕Applying a function rule in the wrong order, e.g. $y = 2x + 1$ with $x = 4$ giving $2 \times (4 + 1) = 10$ .
Why it happens
The order of operations is overlooked when working through a rule.
How to avoid it
Follow BIDMAS: multiply first, then add. $2 \times 4 + 1 = 9$ .
✕Joining graph points with a wobbly freehand line instead of a ruler.
Why it happens
It feels quicker to sketch the line by hand.
How to avoid it
A linear function makes a perfectly straight line — always use a ruler to join the points.
✕Drawing the line through a point that does not line up with the others.
Why it happens
Every plotted point is assumed to be correct.
How to avoid it
If one point is off the line, re-check it in the table — a linear graph has all points in a straight row.

Practice quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Linear functions and Graphs

Detailed Notes

What you’ll learn

1Reading coordinates from a grid

2Using a function rule

3Making a table of values

4Plotting points in four quadrants

5Reading a value from a graph

Coordinates

x-axis

y-axis

Origin

Quadrant

Function

Input

Output

Mapping

Table of values

Linear function

Plot

✕Reading coordinates the wrong way round, e.g. plotting (3,2)(3, 2)(3,2) as 2 across and 3 up.

✕Moving the wrong way for a negative coordinate, e.g. plotting (−2,3)(-2, 3)(−2,3) to the right.

✕Applying a function rule in the wrong order, e.g. y=2x+1y = 2x + 1y=2x+1 with x=4x = 4x=4 giving 2×(4+1)=102 \times (4 + 1) = 102×(4+1)=10.

✕Joining graph points with a wobbly freehand line instead of a ruler.

✕Drawing the line through a point that does not line up with the others.

Practice quiz

✕Reading coordinates the wrong way round, e.g. plotting $(3, 2)$ as 2 across and 3 up.

✕Moving the wrong way for a negative coordinate, e.g. plotting $(-2, 3)$ to the right.

✕Applying a function rule in the wrong order, e.g. $y = 2x + 1$ with $x = 4$ giving $2 \times (4 + 1) = 10$ .