The coordinate plane: points, quadrants and projections on the axes
Mathematics Prep 1 Term 1, Unit 4, Lessons 5–6
GoalPlace points in the coordinate plane, name their quadrant and find their projections on the axes.
New words
quadrant · one of the four regions the axes cut the plane intoorigin · the point (0, 0) where the axes crossprojection · the point of an axis reached by going straight from the point to that axisordered pair · two numbers (x, y) that give the place of a point
Explanation
The x-axis is horizontal and the y-axis is vertical; they cross at the origin (0, 0). A point is written (x, y): first go along the x-axis, then up or down. The axes cut the plane into four quadrants: I has x > 0 and y > 0, II has x < 0 and y > 0, III has x < 0 and y < 0, and IV has x > 0 and y < 0. A point (x, 0) lies on the x-axis and a point (0, y) lies on the y-axis. Points on the axes belong to no quadrant. The projection of a point on the x-axis keeps x and sets y = 0. The projection on the y-axis keeps y and sets x = 0. The length of the projection of a segment on an axis is the positive difference of the coordinates.
Worked examples
Quadrants. A(3, 2): both positive, quadrant I. B(−4, 3): x negative, y positive, quadrant II. C(−2, −3): quadrant III. D(4, −2): quadrant IV. E(5, 0) is on the x-axis and F(0, −3) is on the y-axis, so they are in no quadrant.
Projection of a point. A is (2, 3). Its projection on the x-axis is (2, 0) and on the y-axis is (0, 3).
Projection of a segment. A(1, 4) and B(6, 2). On the x-axis the projection runs from (1, 0) to (6, 0), length |6 − 1| = 5. On the y-axis it runs from (0, 2) to (0, 4), length |4 − 2| = 2.
Area on the grid. A(1, 1), B(5, 1) and C(1, 4). AB lies on the line y = 1 and is 5 − 1 = 4 long. The height is the vertical side AC = 4 − 1 = 3. Area = ½ × 4 × 3 = 6 square units.
An unknown. The point P(2k − 4, 3) is on the y-axis, so its x-coordinate is 0: 2k − 4 = 0, 2k = 4 and k = 2. So P is (0, 3).
Class activity
Treasure map. With a grown-up, draw axes on squared paper from −5 to 5. Hide a treasure at a point and give your partner clues: its quadrant, its projection on the x-axis and its projection on the y-axis.
Practice
1
Name the quadrant: (−5, 2), (−1, −6), (7, −3), (0, 4).
(−5, 2) is in II, (−1, −6) in III, (7, −3) in IV. (0, 4) is on the y-axis, in no quadrant.
2
Find the projections of B(−4, 3) on the x-axis and on the y-axis.
On the x-axis (−4, 0). On the y-axis (0, 3).
3
C(−3, 2) and D(4, −1). Find the lengths of the projections of CD on the two axes.
On the x-axis |4 − (−3)| = 7. On the y-axis |−1 − 2| = 3.
4
The point Q(k + 5, −2) is on the y-axis. Find k and Q.
k + 5 = 0, so k = −5 and Q is (0, −2).
5
Omar says the projection of (2, 3) on the x-axis is (0, 3). What is wrong?
On the x-axis the y-coordinate becomes 0 and x stays: (2, 0). (0, 3) is the projection on the y-axis.
