So far, we have only considered points on the two coordinate axes. What can you say about the coordinates of points that are not on either axes?

Step-by-step solution

Idea: A point on the x-axis has y = 0 and a point on the y-axis has x = 0. A point on neither axis has both coordinates non-zero, and the two signs place it in one of the four quadrants.

1. On the x-axis every point has y = 0, e.g. (4.5, 0). On the y-axis every point has x = 0, e.g. (0, 4). So a point on neither axis has x ≠ 0 and y ≠ 0.½ mark
2. x is the distance of the point from the y-axis: positive if the point is to the right, negative if to the left. y is its distance from the x-axis: positive above, negative below.½ mark
3. The axes divide the plane into four quadrants, and each has its own pair of signs: Quadrant I (+, +), Quadrant II (−, +), Quadrant III (−, −), Quadrant IV (+, −).1 mark
4. For example, S (3, −5) has x positive and y negative, so it is in Quadrant IV; Q (−5, 3) is in Quadrant II.

Check: (0, 0) is on both axes and fits neither quadrant; any point with a zero coordinate is on an axis, never inside a quadrant ✓.

Answer to write in the exam

Common mistakes that cost marks

• Numbering the quadrants clockwise. They go anticlockwise from the top right: I, II, III, IV.
• Saying a point like (0, −3) is in Quadrant IV. A zero coordinate puts it on an axis, not in any quadrant.
• Mixing up the signs of Quadrants II and IV: in II the x-coordinate is negative; in IV the y-coordinate is negative.

How this can come in the exam

Try one yourself

Say where each point lies (which quadrant, or which axis): (−3, −8), (0, −6), (5, 2), (−1, 0.5).

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