Linear Graphs and Coordinate Geometry: Year 8 Study Guide
A straight-line graph shows how one quantity changes steadily with another.
The Cartesian plane has an x-axis, a y-axis and four quadrants. A linear relationship has a constant rate of change and graphs as a straight line. The gradient (slope) = rise ÷ run, and the y-intercept is where the line crosses the y-axis. A rule such as y = 3x − 2 has a gradient of 3 and a y-intercept of −2.
To graph a line, make a table of values: for y = 3x − 2, x = 0, 1, 2 gives y = −2, 1, 4. Plot the points and join them with a ruler. Two lines are parallel if they have the same gradient.
Gradient and intercept
| Rule | Gradient | y-intercept | Direction |
|---|---|---|---|
| y = 2x + 1 | 2 | 1 | rises |
| y = −x + 4 | −1 | 4 | falls |
| y = 5 | 0 | 5 | horizontal |
| y = ½x − 3 | ½ | −3 | rises gently |
Real-world graphs
A mobile plan costs $10 a month plus $0.05 per megabyte over the limit: C = 0.05m + 10. The gradient 0.05 is the rate per MB and 10 is the starting cost. Reading from the graph answers questions: for $25, 0.05m = 15, m = 300 MB. Check the axes for units and scales before reading values. Distance–time graphs: a steeper line means faster; a horizontal line means stopped.
Choose x values.
On a grid.
With a ruler, extend it.
Interpret the gradient.
Midpoint and distance
The midpoint of (2, 3) and (8, 7) is ((2 + 8) ÷ 2, (3 + 7) ÷ 2) = (5, 5). The distance between them uses Pythagoras: √(6² + 4²) = √52 ≈ 7.2 units. Two points with the same x-coordinate lie on a vertical line, and with the same y-coordinate on a horizontal line.
Worked examples and common mistakes
Example: Find the gradient through (1, 2) and (4, 11): (11 − 2) ÷ (4 − 1) = 3. Common mistakes: swapping x and y, reading the scale incorrectly and finding run ÷ rise. Always label the axes and units.
Check your understanding: model answers
1. 9 ÷ 3 = 3. 2. −6. 3. (6, 4).
✨ Key Takeaways
- Gradient = rise ÷ run.
- y = mx + c: m is the gradient and c the y-intercept.
- Use a table of values to plot.
- Midpoint = average of coordinates.
💬 Questions for Discussion & Reflection
- Find the gradient through (0, 1) and (3, 10).
- State the y-intercept of y = 4x − 6.
- Find the midpoint of (2, 6) and (10, 2).
Print coordinate plane grids →