IGCSE Linear Graphs and Coordinate Geometry
Read and build straight-line equations, then use coordinates to find midpoints and distances.
🎯 Key idea: In $y=mx+c$, $m$ tells you the vertical change for every one step right; $c$ tells you where the line crosses the $y$-axis.
📈 1. Gradient and equations
A positive gradient rises left to right; a negative gradient falls. Use $m=[[frac{change in y}{change in x}]]$.
Worked example: A line rises $6$ when it moves $3$ right.
$m=[[frac{6}{3}]]=2$.
If it crosses the $y$-axis at $-4$, its equation is $y=2x-4$.
Worked example: Find the equation with gradient $3$ through $(2,1)$.
Use $y=3x+c$ and substitute the point.
$1=3(2)+c$, so $c=-5$.
The equation is $y=3x-5$.
🎯 2. Midpoint
The midpoint is the average of the two $x$-coordinates and the average of the two $y$-coordinates.
Worked example: Find the midpoint of $(2,-3)$ and $(8,5)$.
$x=[[frac{2+8}{2}]]=5$ and $y=[[frac{-3+5}{2}]]=1$.
The midpoint is $(5,1)$.
📏 3. Distance on a grid
For horizontal or vertical segments, subtract the matching coordinates. For other segments, use Pythagoras.
Worked example: Find the distance from $(1,2)$ to $(7,10)$.
The horizontal and vertical changes are $6$ and $8$.
Distance $=[[sqrt{6^2+8^2}]]=10$.
Graph explorer: Before drawing, predict: what changes if only $m$ changes? What changes if only $c$ changes? Test three equations with the same $m$ and different $c$ values on graph paper or a graphing app.
Practice Problems
Sketch or calculate before opening each answer.
1. Find the gradient through $(1,4)$ and $(5,12)$.
Show answer
$m=[[frac{12-4}{5-1}]]=2$.
Answer: $2$.
2. Find the midpoint of $(-4,7)$ and $(6,1)$.
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$([[frac{-4+6}{2}]],[[frac{7+1}{2}]])=(1,4)$.
Answer: $(1,4)$.
3. Find the distance from $(0,0)$ to $(3,4)$.
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$[[sqrt{3^2+4^2}]]=5$.
Answer: $5$ units.
Interactive Quiz
Choose an answer, then submit the quiz.
1. In $y=-2x+7$, what is the gradient?
2. What is the midpoint of $(2,1)$ and $(6,9)$?
3. What is the distance from $(1,1)$ to $(1,9)$?
Linear Graph Explorer
Draw lines from gradient and intercept, or type an equation. Compare multiple lines on the same grid.
Add a line to see it on the grid.