Point W has x-coordinate equal to −5. Can you predict the coordinates of point H which is on the line through W parallel to the y-axis? Which quadrants can H lie in?
Step-by-step solution
Idea: A line parallel to the y-axis is a vertical line. Every point on it is the same distance from the y-axis, so every point has the same x-coordinate.
- The line through W parallel to the y-axis is vertical. Every point on it is 5 units to the left of the y-axis, so every point has x-coordinate −5.½ mark
- Its y-coordinate can be any number. So H = (−5, k), where k is any number, for example (−5, 3) or (−5, −2).½ mark
- The x-coordinate is negative, so H is to the left of the y-axis: in Quadrant II when k > 0 and in Quadrant III when k < 0.½ mark
- When k = 0, H = (−5, 0) lies on the x-axis and is in no quadrant. H can never be in Quadrant I or IV.½ mark
Check: (−5, 3): signs (−, +), Quadrant II ✓. (−5, −2): signs (−, −), Quadrant III ✓.
Answer to write in the exam
Line through W parallel to the y-axis: every point has x = −5
∴ H = (−5, k), k any real number
k > 0 ⇒ H in Quadrant II; k < 0 ⇒ H in Quadrant III; k = 0 ⇒ H = (−5, 0) on the x-axis
∴ H can lie in Quadrant II or Quadrant III.
Common mistakes that cost marks
- Giving H = (k, −5). A line parallel to the y-axis keeps the x-coordinate fixed, not the y-coordinate.
- Saying H can be in all four quadrants. Its x-coordinate is always −5, so it is always to the left of the y-axis.
- Forgetting the special case (−5, 0), which lies on the x-axis, not in a quadrant.
How this can come in the exam
All points on the line through (3, −2) parallel to the y-axis have
- x-coordinate 3
- y-coordinate −2
- x-coordinate −2
- y-coordinate 3
Show answer
(A) x-coordinate 3
A line parallel to the y-axis is vertical, so the x-coordinate stays 3.
A line is drawn through P (4, 6) parallel to the x-axis. Write the general form of a point on it, one point on it in Quadrant II, and the point where it meets the y-axis.
Show answer
Every point has y = 6: (k, 6) (½ mark). In Quadrant II, e.g. (−3, 6) (½ mark). It meets the y-axis where x = 0: (0, 6) (1 mark).Try one yourself
Point K has y-coordinate −3. L is on the line through K parallel to the x-axis. Write L’s coordinates and the quadrants L can lie in.
Show answer
L = (k, −3). L is in Quadrant IV if k > 0, Quadrant III if k < 0, and on the y-axis at (0, −3) if k = 0.
More questions like this
- Consider the points R (3, 0), A (0, −2), M (−5, −2) and P (−5, 2). If they are joined in the same order, predict:
- Plot point Z (5, −6) on the Cartesian plane. Construct a right-angled triangle IZN and find the lengths of the three sides.
(Comment: Answers may differ from person to person.) - What would a system of coordinates be like if we did not have negative numbers? Would this system allow us to locate all the points on a 2-D plane?
- Are the points M (−3, −4), A (0, 0) and G (6, 8) on the same straight line? Suggest a method to check this without plotting and joining the points.
- Use your method (the one used for M (−3, −4), A (0, 0) and G (6, 8)) to check if the points R (−5, −1), B (−2, −5) and C (4, −12) are on the same straight line. Now plot both sets of points and check your answers.