Coordinates, Linear Graphs & Gradient

Q: How do I remember which coordinate comes first?

Coordinates are written as : the horizontal x-value first, then the vertical y-value, like walking along a corridor then up the stairs.

Q: Is gradient rise over run or run over rise?

Gradient is always rise over run: vertical change divided by horizontal change between two points on the line.

Mathematics· 3.1, 3.2, 3.3, E3.4· 25 min read

1. Core: Coordinates and Plotting Linear Graphs★★☆☆☆⏱ 6 min

All points on a Cartesian grid are written as $(x, y)$ , where $x$ is the horizontal axis value (left/right from the origin $(0, 0)$ ) and $y$ is the vertical axis value (up/down from the origin). Linear graphs are straight lines that follow the equation $y = m x + c$ , where $m$ is the gradient and $c$ is the y-intercept (the point where the line crosses the y-axis).

📘 Definition

Linear Graph

A straight-line graph that represents a relationship between two variables with a constant rate of change (gradient), written in the form $y = m x + c$ for Core content.

Example:

$y = 2 x + 3$ is a linear graph with gradient 2 and y-intercept 3.

📐 Worked Example

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

1
Create a table of values: for $x = - 2$ , $y = 2 (- 2) + 1 = - 3$ ; $x = - 1$ , $y = - 1$ ; $x = 0$ , $y = 1$ ; $x = 1$ , $y = 3$ ; $x = 2$ , $y = 5$
2
Plot each point $(x, y)$ on a Cartesian grid: $(- 2, - 3)$ , $(- 1, - 1)$ , $(0, 1)$ , $(1, 3)$ , $(2, 5)$
3
Draw a straight line through all plotted points, extending it slightly past the outermost points

2. Core: Calculating Gradient from a Grid★★☆☆☆⏱ 6 min

Gradient measures the steepness of a line. A positive gradient means the line slopes upwards from left to right, a negative gradient means it slopes downwards, and a gradient of 0 means the line is horizontal. For Core exams, you calculate gradient by counting squares on a grid: it is the vertical change (rise) divided by the horizontal change (run) between two clear points on the line.

📘 Definition

Gradient (Core)

The steepness of a line, calculated as $gradient = \frac{vertical change (rise)}{horizontal change (run)}$ by counting squares on a grid.

Example:

A line that rises 4 squares for every 2 squares it runs to the right has a gradient of $4/2 = 2$ .

📐 Worked Example

Calculate the gradient of the line on the grid that passes through points $(0, 1)$ and $(2, 5)$ .

1
Find the vertical change: from $y = 1$ to $y = 5$ , the rise is $5 - 1 = + 4$
2
Find the horizontal change: from $x = 0$ to $x = 2$ , the run is $2 - 0 = + 2$
3
Divide rise by run: gradient = $4/2 = 2$

3. Extended: Gradient from Coordinates & $ax + by = c$ Form★★★☆☆Extended only⏱ 7 min

For Extended exams, you do not need a grid to calculate gradient: you can use the coordinates of any two points on the line. You also need to work with linear equations written in the form $a x + b y = c$ , which you can rearrange to $y = m x + c$ to find the gradient and intercept.

📘 Definition

Gradient (Extended)

The steepness of a line, calculated from two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ as $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ .

📐 Worked Example

Calculate the gradient of the line passing through points $(3, 4)$ and $(- 1, - 2)$ .

1
Label the points: $x_{1} = 3, y_{1} = 4, x_{2} = - 1, y_{2} = - 2$
2
$m = \frac{-2 - 4}{-1 - 3} = \frac{-6}{-4} = \frac{3}{2}$

📐 Worked Example

Find the gradient and y-intercept of the line $2 x + 3 y = 12$ .

1
Rearrange the equation to the form $y = m x + c$ : subtract $2 x$ from both sides: $3 y = - 2 x + 12$
2
Divide all terms by 3: $y = - \frac{2}{3} x + 4$
3
Identify gradient $m = - \frac{2}{3}$ and y-intercept $c = 4$

4. Extended: Midpoint & Length of Line Segments★★★☆☆Extended only⏱ 6 min

Extended candidates also need to calculate the midpoint (the point exactly halfway between two endpoints of a line segment) and the total length of a line segment from its endpoints.

📘 Definition

Midpoint of a Line Segment

The point halfway between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , calculated as $M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$ .

📘 Definition

Length of a Line Segment

The distance between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , calculated using Pythagoras' theorem: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ .

📐 Worked Example

Find the midpoint and length of the line segment joining points $(2, 3)$ and $(6, 7)$ .

1
Calculate midpoint: $M = (\frac{2 + 6}{2}, \frac{3 + 7}{2}) = (4, 5)$
2
Calculate length: first find x difference: $6 - 2 = 4$ , y difference: $7 - 3 = 4$
3
$d = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$

5. Common Pitfalls

Wrong move:

Swapping x and y coordinates when plotting points

Why:

Coordinates are ordered $(x, y)$ , so swapping them will plot the point in the wrong position, leading to incorrect graphs or gradient calculations.

Correct move:

Always remember 'x is across, y is up': write the horizontal value first, then the vertical value.

Wrong move:

Calculating run over rise instead of rise over run for gradient

Why:

This gives the inverse of the correct gradient, which will be marked wrong even if the slope direction is right.

Correct move:

Use the mnemonic 'rise before run': divide vertical change by horizontal change, or use the formula $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ for Extended.

Wrong move:

Forgetting that a downward sloping line has a negative gradient

Why:

Counting rise as a positive number for a line that slopes down will give the wrong sign for gradient.

Correct move:

If the line goes down as you move right, your vertical change is negative, so your gradient will be negative.

Wrong move:

Mixing up the order of x or y values in the gradient or distance formula (Extended only)

Why:

If you do $y_{1} - y_{2}$ for the numerator and $x_{2} - x_{1}$ for the denominator, you will get the wrong sign for gradient.

Correct move:

Always subtract the coordinates of the first point from the second point for both x and y values.

Wrong move:

Forgetting to square the differences in the line length formula (Extended only)

Why:

This is a common arithmetic error that leads to an incorrect distance value.

Correct move:

Always square both the x difference and y difference before adding them and taking the square root.

6. Quick Reference Cheatsheet

Concept	Core Requirement	Extended Requirement
Plotting coordinates	Read and plot $(x, y)$ points on a grid	Same as Core
Linear equation form	Use $y = m x + c$ , identify $m$ and $c$	Rearrange $a x + b y = c$ to $y = m x + c$
Gradient calculation	Count rise/run from grid squares	Use formula $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ from coordinates
Line segment properties	Not assessed	Midpoint: $(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$ , Length: $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

7. Frequently Asked

How do I remember which coordinate comes first?

Coordinates are written as $(x, y)$ : the horizontal x-value first, then the vertical y-value, like walking along a corridor then up the stairs.

Is gradient rise over run or run over rise?

Gradient is always rise over run: vertical change divided by horizontal change between two points on the line.

Do I need to simplify gradient values in exams?

Yes, always write gradients in their simplest fractional or integer form, unless the question specifies otherwise.

Going deeper

syllabusCIE IGCSE Maths 0580 2025-2027 SyllabusOfficial specification for this topic

What's Next

Now that you have mastered coordinates, linear graphs, and gradient for CIE IGCSE Maths 0580, you are ready to move on to more advanced coordinate geometry topics. The next sub-topic covers equations of parallel and perpendicular lines, which builds directly on your understanding of gradient and linear equation forms. You can also practice applying these skills to past paper questions to reinforce your learning and identify any gaps before your exam. Make sure you are comfortable with both Core and Extended content relevant to your tier before progressing to more complex topics.