📈 Linear Graphs

Grade 8 · Cambridge Lower Secondary Stage 8

What you'll learn

  • Plot straight-line graphs from equations in the form y = mx + c
  • Identify gradient (m) and y-intercept (c)
  • Find the equation of a line from two points or a graph
  • Understand parallel and perpendicular lines
  • Find where two lines intersect algebraically

📖 Learn

1. y = mx + c

Every straight line can be written as y = mx + c where:

y = mx + c
m = gradient (steepness). Positive → slopes up. Negative → slopes down.
c = y-intercept (where the line crosses the y-axis).
Example: y = 3x − 2 has gradient 3 and y-intercept −2
Special cases: y = 4 is a horizontal line (m = 0). x = 3 is a vertical line.
💡 To plot: mark the y-intercept, then use gradient (rise ÷ run) to find the next point.

2. Gradient

Gradient = how steep the line is = change in y ÷ change in x.

m = (y₂ − y₁) ÷ (x₂ − x₁)
Positive gradient: line rises left to right
Negative gradient: line falls left to right
Zero gradient: horizontal line
Example: Points (1, 3) and (4, 9): m = (9−3)÷(4−1) = 6÷3 = 2
💡 Rise over Run: gradient = vertical change ÷ horizontal change between any two points.

3. Finding the Equation of a Line

Given two points, you can find the equation y = mx + c:

Step 1: Calculate gradient m = (y₂−y₁)÷(x₂−x₁)
Step 2: Use one point and y = mx + c to find c
Step 3: Write the full equation
Example: Through (2, 5) and (4, 11): m = (11−5)÷(4−2) = 3. Use (2,5): 5 = 3(2)+c → c = −1. Equation: y = 3x − 1

4. Parallel and Perpendicular Lines

Parallel lines have the same gradient but different y-intercepts.

Parallel to y = 2x + 5: any line y = 2x + c (same m = 2)

Perpendicular lines have gradients that multiply to −1.

m₁ × m₂ = −1    so    m₂ = −1/m₁
Perpendicular to y = 3x + 1: gradient = −1/3
⚠️ Perpendicular: flip the fraction and change the sign (negative reciprocal).

5. Intersection of Two Lines

Two lines meet at one point (unless parallel). Find intersection by solving simultaneously.

Example: y = 2x + 1 and y = x + 3
Set equal: 2x + 1 = x + 3 → x = 2
Substitute: y = 2(2) + 1 = 5
Intersection: (2, 5)
💡 This is the same as solving simultaneous equations — the intersection IS the solution.

✏️ Worked Examples

Example 1 – Reading Gradient and Intercept

State the gradient and y-intercept of: (a) y = 4x − 3   (b) y = −2x + 7   (c) 2y = 6x + 10

(a) y = 4x − 3 → m = 4, c = −3
(b) y = −2x + 7 → m = −2, c = 7
(c) Divide by 2: y = 3x + 5 → m = 3, c = 5

Example 2 – Gradient from Two Points

Find the gradient of the line through (1, 4) and (5, 12).

Step 1: m = (12 − 4) ÷ (5 − 1) = 8 ÷ 4 = 2
Check: Every 1 step right, go 2 up — confirms gradient = 2 ✓

Example 3 – Finding the Equation

Find the equation of the line through (3, 7) and (6, 13).

Step 1: m = (13 − 7) ÷ (6 − 3) = 6 ÷ 3 = 2
Step 2: y = 2x + c. Use (3, 7): 7 = 2(3) + c → 7 = 6 + c → c = 1
Equation: y = 2x + 1
Check: At x = 6: y = 2(6)+1 = 13 ✓

Example 4 – Parallel and Perpendicular

Line L: y = 4x + 1. Find the equation of: (a) a parallel line through (0, −3)   (b) a perpendicular line through (0, 2)

(a) Parallel: same gradient m = 4. Through (0, −3) means c = −3. Equation: y = 4x − 3
(b) Perpendicular: gradient = −1/4. Through (0, 2) means c = 2. Equation: y = −¼x + 2

🎨 Visualizer

📈 Line Plotter

Plot up to two lines y = mx + c on the same axes.

📐 Gradient from Two Points

🔲 Parallel & Perpendicular Checker

Enter gradients of two lines. Find out their relationship.

❌ Intersection Finder

y = m₁x + c₁ and y = m₂x + c₂. Find where they meet.

Exercise 1 – Gradient and Y-intercept

For y = mx + c, give the gradient (m). Even-numbered questions ask for c instead — check question carefully.

Exercise 2 – Gradient from Two Points

Calculate the gradient of the line through the given points.

Exercise 3 – Finding the Y-intercept

You are given the gradient and a point on the line. Find the y-intercept c.

Exercise 4 – Parallel and Perpendicular Gradients

Enter the gradient of the parallel or perpendicular line as directed. Use decimals (e.g. −0.5).

Exercise 5 – Intersection of Two Lines

Odd questions: give the x-coordinate of intersection. Even questions: give the y-coordinate.

📝 Practice – 20 Questions

🏆 Challenge – 8 Questions