Plotting points on a graph

Two number lines cross at right angles at the origin, (0, 0). The horizontal one is the x-axis and the vertical one is the y-axis. Every point in the plane is described by an ordered pair (x, y): its horizontal position and its vertical position.

The axes split the plane into four quadrants, numbered anticlockwise from the top right: I (x > 0, y > 0), II (x < 0, y > 0), III (both negative) and IV (x > 0, y < 0). Points on an axis are in no quadrant.

1. Start at the origin.
2. Move along the x-axis by the first number: right if it is positive, left if it is negative.
3. From there, move parallel to the y-axis by the second number: up if positive, down if negative.
4. Mark the point and label it if needed.

(3, 2) and (2, 3) are different points. The pair is ordered: x always comes first. A useful memory aid is that you walk along the corridor before you go up the stairs.

To draw y = x² − 2 by hand, choose some x-values, compute y for each, and plot the pairs:

Then join the points with a smooth curve, not straight segments, since the function is defined for every x in between. In the calculator, add a table from the + menu, type the values, and the points appear; type y = x^2 - 2 on another row to see the curve go through them.

Two plotted points give two classic formulas. The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²), which is Pythagoras’ theorem. The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). For (−2, −3) and (3, 2) the distance is √(25 + 25) = 5√2 ≈ 7.07 and the midpoint is (0.5, −0.5).