7.2 The Cartesian Plane & Linear Graphs
Key Takeaways
- An ordered pair (x, y) moves horizontally then vertically from the origin; (3, -2) and (-2, 3) are different points
- Quadrants run anticlockwise from the top right: (+,+), (-,+), (-,-), (+,-); a point with a zero coordinate lies on an axis, not in a quadrant
- Vertical lines have equation x = k and horizontal lines y = k; the x-axis itself is the line y = 0
- In y = mx + c, m is the gradient (rise ÷ run) and c is the y-intercept, the point (0, c) where the line crosses the y-axis
- On a distance-time graph the gradient equals speed, a flat section means stopped, and a section sloping back towards the time axis means returning
The Cartesian plane appears from Paper D (reading plotted points) through Papers E-F, where sketching and interpreting straight-line graphs is a core skill, and into the upper papers where graphs are used to solve equations. ICAS loves questions that display a graph and ask you to read information from it, so fluency with coordinates pays off across several questions in every paper.
Coordinates and the Plane
The Cartesian plane is formed by two number lines crossing at right angles: the horizontal x-axis and the vertical y-axis. They meet at the origin, the point (0, 0). Every point is described by an ordered pair (x, y): the x-value tells you how far to move horizontally (right is positive) and the y-value how far to move vertically (up is positive).
To plot (3, -2): start at the origin, move 3 units right, then 2 units down. The order matters — (3, -2) and (-2, 3) are completely different points.
The Four Quadrants
The axes divide the plane into four quadrants, numbered anticlockwise starting from the top right.
| Quadrant | Sign of x | Sign of y | Example point |
|---|---|---|---|
| I | + | + | (2, 5) |
| II | - | + | (-4, 3) |
| III | - | - | (-1, -6) |
| IV | + | - | (5, -3) |
A point with a zero coordinate, such as (0, 4) or (-3, 0), lies on an axis, not in any quadrant — a favourite ICAS trick option.
Horizontal and Vertical Lines
- Every point on a vertical line has the same x-value, so its equation is x = k; for example, x = 3 passes through (3, 0), (3, 5) and (3, -7).
- Every point on a horizontal line has the same y-value, so its equation is y = k; for example, y = -2.
- The x-axis itself is the line y = 0, and the y-axis is x = 0.
If a question asks which line passes through (2, 5) and (2, -1), notice that both points have x = 2, so the answer is the vertical line x = 2 — no calculation needed.
From a Rule to a Graph: Tables of Values
A linear rule such as y = 2x + 1 pairs every x-value with a y-value. A table of values organises a few of these pairs; plotting them always produces a straight line.
For y = 2x + 1:
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | -1 | 1 | 3 | 5 |
Choose easy x-values (0, 1, 2 and one negative), compute y mentally, plot the points and draw the line through them with a ruler.
Gradient and y-intercept
In y = mx + c:
- m is the gradient — how steep the line is. It equals rise ÷ run: for every 1 unit moved right, the line rises m units. A positive gradient slopes up to the right, a negative gradient slopes down, and m = 0 gives a horizontal line.
- c is the y-intercept — the y-value where the line crosses the y-axis, the point (0, c).
Example. For y = 3x - 2, the gradient is 3 and the line cuts the y-axis at (0, -2).
You can also find the gradient from two points: m = (change in y) ÷ (change in x). Through (1, 2) and (3, 8): m = (8 - 2)/(3 - 1) = 6/2 = 3. Estimation check: the line climbs 6 while running 2, so a steep positive answer is right.
Reading Graphs to Solve Equations
Papers E-F ask you to solve equations graphically. To solve 2x + 1 = 5 from the graph of y = 2x + 1, find where the line reaches y = 5 and read the x-value underneath: x = 2. Two plotted lines solve a pair of equations the same way — read the intersection point.
Distance-Time Graphs
A distance-time graph plots distance travelled against time, and ICAS uses these to test interpretation rather than calculation.
- The gradient equals speed: steeper means faster.
- A horizontal section means stopped — time passes but the distance is unchanged.
- A straight sloping section means constant speed; a curved section means speeding up or slowing down.
- A section sloping back towards the time axis means returning towards the starting point.
Example. A cyclist's graph climbs from 0 to 10 km over the first 2 hours (5 km/h), is flat for the next hour (resting), then falls back to 0 km in 30 minutes. The steepest stage is the ride home: 10 km in half an hour is 20 km/h, far faster than the outward ride.
Common Traps
- Swapping the coordinates: (4, 1) is not the same point as (1, 4).
- Reading run ÷ rise instead of rise ÷ run for a gradient, or mixing up the y-intercept with the x-intercept.
- Forgetting that a negative gradient slopes downwards from left to right.
- In distance-time questions, dividing by the total time instead of the moving time when finding a speed.
Which of the following points lies in the third quadrant of the Cartesian plane?
A straight line passes through the points (0, 3) and (2, 7). What is the gradient of the line?