Printable · GCSE Foundation · ages 14-16
Coordinates in all four quadrants worksheet — GCSE Foundation
Fifteen questions on "coordinates in all four quadrants" — DfE statement A8. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Coordinates in all four quadrants worksheet — GCSE Foundation
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- 1.The point (−4, 7) is moved 4 units to the right and 9 units down. Write down the coordinates of the point it reaches.
- 2.Work out the coordinates of the point halfway between (−9, −2) and (−1, −2).
- 3.The point C is 6 units to the right of the origin and 0 units up. Write down the coordinates of C.
- 4.On a map, where each grid unit represents 1 km, Ravi's house is at (−3, 4) and the library is at (6, 4). Work out the distance between Ravi's house and the library.
- 5.ABCD is a parallelogram. A has coordinates (−3, 1), B has coordinates (2, 1) and C has coordinates (4, 4). Work out the coordinates of D.
- 6.A square has vertices at (−2, 3), (−2, −3) and (4, −3). Write down the coordinates of the fourth vertex.
- 7.Work out the distance of the point (−6, 4) from the x-axis.
- 8.A point lies on the y-axis. What can be said about the x-coordinate of that point?
- 9.Write down the coordinates of the point that is 4 units to the left of the origin and 7 units up.
- 10.Point R has coordinates (3, −5). Point R is reflected in the line y = 2 to point S. Write down the coordinates of S.
- 11.In which quadrant does the point (−3, 0.5) lie?
- 12.A line segment has one endpoint at (−6, 2) and its midpoint at (−1, 5). Work out the coordinates of the other endpoint.
- 13.Which of these points lies in the first quadrant?
- 14.Point P has coordinates (5, 2). Point P is rotated 180° about the origin to point Q. Write down the coordinates of Q.
- 15.The point F has x-coordinate −1 and lies 6 units above the x-axis. Write down the coordinates of F.
Answer key
- (b) (0, −2) — Method: a translation acts on the two coordinates separately: moving right or left changes the x-coordinate only, moving up or down changes the y-coordinate only, and right and up add while left and down subtract. Working: the point starts at (−4, 7); moving 4 units to the right gives an x-coordinate of −4 + 4 = 0; moving 9 units down gives a y-coordinate of 7 − 9 = −2. Answer: (0, −2). The distractors: (5, 3) comes from pairing each number with the wrong coordinate, adding 9 to −4 and taking 4 from 7; (0, 16) comes from treating 'down' as an addition, giving 7 + 9 = 16 for the second coordinate; (−8, −2) comes from treating 'to the right' as a subtraction, giving −4 − 4 = −8 for the first coordinate.
- (b) (−5, −2) — Both points have the same y-coordinate, so the midpoint lies on the same horizontal line: y = −2. The x-coordinate is the average of −9 and −1: (−9 + (−1)) ÷ 2 = −10 ÷ 2 = −5, giving (−5, −2). (−10, −2) comes from adding the x-coordinates but forgetting to divide by 2. (−4, −2) comes from a sign error on the second x-coordinate, treating −1 as +1: (−9 + 1) ÷ 2 = −4. (5, −2) comes from dropping the negative sign on the x-coordinate.
- (a) (6, 0) — Method: a pair of coordinates records movement from the origin, the across movement written first and the up or down movement second. Working: C is 6 units to the right of the origin, so the across number is 6; it is 0 units up, so the up number is 0; written in order that gives (6, 0), a point on the x-axis. Answer: (6, 0). The distractors: (0, 6) comes from writing the two movements the wrong way round; (0, 0) comes from reading '0 units up' as meaning the point never left the origin at all, which ignores the movement across; (7, 0) comes from counting the origin itself as the first unit while counting 6 units to the right.
- (a) 9 km — Ravi's house and the library have the same y-coordinate, so the distance between them is the difference between their x-coordinates: 6 − (−3) = 9 km. 3 km comes from dropping the negative sign on Ravi's x-coordinate and working out 6 − 3 instead. 0 km comes from using the y-coordinates, which are equal, instead of the x-coordinates. −9 km comes from working out −3 − 6 = −9 and not converting it to a positive distance.
- (d) (−1, 4) — In parallelogram ABCD the side DC is parallel and equal to the side AB, so D = C − AB. The vector from A to B is (2 − (−3), 1 − 1) = (5, 0), so D = (4 − 5, 4 − 0) = (−1, 4). A candidate who adds this vector to C instead of subtracting it gets (4 + 5, 4 + 0) = (9, 4). A candidate who subtracts A's coordinates from C's rather than the vector AB, and drops the minus sign on −3 while doing so, works out (4 − 3, 4 − 1) and gets (1, 3). A candidate who makes only the y-part of that slip, working out 4 − 1 instead of 4 − 0, gets (−1, 3).
- (d) (4, 3) — The sides are parallel to the axes: the missing vertex must share the y-coordinate 3 with (−2, 3) and the x-coordinate 4 with (4, −3), giving (4, 3). (−4, 3) comes from a sign error on the x-coordinate. (4, −9) comes from continuing the pattern of the given points by subtracting 6 from the y-coordinate again instead of matching it to (−2, 3). (3, 4) comes from swapping the x- and y-coordinates.
- (c) 4 — Method: the distance of a point from the x-axis is measured vertically, so it is the size of the y-coordinate taken without its sign. Working: the point (−6, 4) has y-coordinate 4, so moving straight down to the x-axis covers 4 units, and a distance is written as a positive number. Answer: 4. The distractors: 6 comes from using the x-coordinate, which measures the distance from the y-axis rather than from the x-axis; −6 comes from that same mistake with the minus sign left in place, although a distance is never negative; 10 comes from adding the two distances, 6 and 4, as though the question asked how far the point is from both axes together.
- (b) x = 0 — Method: the first coordinate of a point measures how far to the left or right of the origin it is, so a point that is neither left nor right of the origin has a first coordinate of zero. Working: the y-axis is the vertical line through the origin; every point on it lies directly above or directly below the origin with no sideways movement at all, so its first coordinate is zero while its second coordinate may take any value. Answer: x = 0. The distractors: y = 0 is the condition for lying on the x-axis, the other axis; x = 1 comes from confusing the y-axis with the vertical line one unit to the right of it; x = y is the condition for the diagonal through the origin, which meets the y-axis at the origin only.
- (c) (−4, 7) — Left of the origin means negative x, and up means positive y, so the point is (−4, 7). (4, 7) comes from forgetting that 'left' means the x-coordinate is negative. (−4, −7) comes from treating 'up' as a negative direction instead of positive. (7, −4) comes from swapping the x- and y-coordinates.
- (c) (3, 9) — Reflecting in the horizontal line y = 2 keeps the x-coordinate the same and maps y to 2 × 2 − y = 4 − (−5) = 9, so S = (3, 9). A candidate who uses k − y instead of 2k − y gets 2 − (−5) = 7, giving (3, 7). A candidate who reflects in the x-axis instead of the line y = 2, simply changing the sign of y, gets (3, 5). A candidate who also changes the sign of the x-coordinate, as if reflecting in both axes, gets (−3, 9).
- (a) the second quadrant — Method: the quadrant is settled by the signs of the two coordinates, not by their size; the quadrants are numbered anticlockwise, starting from the region where both coordinates are positive. Working: the x-coordinate −3 is negative, so the point lies to the left of the y-axis; the y-coordinate 0.5 is positive, so it lies above the x-axis; the region that is both left of the y-axis and above the x-axis is the second. Answer: the second quadrant. The distractors: 'the first quadrant' comes from ignoring the minus sign on −3; 'the third quadrant' comes from treating 0.5 as a negative value because it is smaller than 1, when in fact any number above zero is positive; 'the fourth quadrant' comes from reading the pair the wrong way round, as though the point were (0.5, −3).
- (d) (4, 8) — The other endpoint is found from 2 × midpoint − known endpoint: x = 2 × (−1) − (−6) = −2 + 6 = 4, y = 2 × 5 − 2 = 10 − 2 = 8, giving (4, 8). (−3.5, 3.5) comes from averaging the given endpoint and the midpoint as if they were the two endpoints of a segment, instead of working backwards from the midpoint. (5, 3) comes from working out (−1 − (−6), 5 − 2) instead of doubling the midpoint before subtracting. (4, 5) comes from correctly finding the x-coordinate but copying the midpoint's y-coordinate of 5 instead of doubling it.
- (d) (3, 5) — Method: the quadrant a point lies in is decided by the signs of its two coordinates, and in the first quadrant both coordinates are positive. Working: (−2, 3) has a negative x-coordinate, so it sits to the left of the y-axis; (4, −1) has a negative y-coordinate, so it sits below the x-axis; (−1, −4) has both coordinates negative; only (3, 5) has a positive x-coordinate and a positive y-coordinate. Answer: (3, 5). The distractors: (−2, 3) is chosen by candidates who check only the y-coordinate and take a positive height as enough; (4, −1) is chosen by those who check only the x-coordinate; (−1, −4) is chosen by those who number the quadrants from the bottom left, so that the region with two negative coordinates is called the first.
- (b) (−5, −2) — A rotation of 180° about the origin reverses the sign of both coordinates, so Q = (−5, −2). A candidate who reverses the sign of only the y-coordinate, as if reflecting in the x-axis, gets (5, −2). A candidate who reverses the sign of only the x-coordinate, as if reflecting in the y-axis, gets (−5, 2). A candidate who swaps the coordinates instead of reversing their signs gets (2, 5).
- (b) (−1, 6) — '6 units above the x-axis' gives a y-coordinate of 6, and the x-coordinate is given as −1, so F = (−1, 6). (6, −1) comes from swapping the x- and y-coordinates. (−1, −6) comes from treating 'above' as a negative direction instead of positive. (1, 6) comes from dropping the negative sign on the given x-coordinate.
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