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
MathsUKwww.geekhero.co.uk
- 1.Work out the distance of the point (−6, 4) from the x-axis.
- 2.Point R has coordinates (3, −5). Point R is reflected in the line y = 2 to point S. Write down the coordinates of S.
- 3.The point B has coordinates (−2, −2). In which quadrant is B?
- 4.A point has coordinates (x, y). In which two quadrants is the product x × y positive?
- 5.Point P has coordinates (2, −5). P is reflected in the x-axis to point Q. Write down the coordinates of Q.
- 6.A triangle has vertices at (1, 1), (1, 5) and (6, 1). Work out the area of the triangle.
- 7.Work out the coordinates of the reflection of (6, −3) in the y-axis.
- 8.The point F has x-coordinate −1 and lies 6 units above the x-axis. Write down the coordinates of F.
- 9.Which of these points lies in the first quadrant?
- 10.A line segment has one endpoint at (−6, 2) and its midpoint at (−1, 5). Work out the coordinates of the other endpoint.
- 11.A point lies on the y-axis. What can be said about the x-coordinate of that point?
- 12.The point P has coordinates (0, 12). Write down the axis that P lies on and the y-coordinate of P.
- 13.Three vertices of a rectangle are (−4, −1), (2, −1) and (2, 3). The sides of the rectangle are parallel to the axes. Write down the coordinates of the fourth vertex.
- 14.The point A lies on the x-axis. Which statement about A must be true?
- 15.Write down the coordinates of the point that is 4 units to the left of the origin and 7 units up.
Answer key
- (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.
- (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).
- (d) the third quadrant — Method: only the signs of the coordinates matter, so two coordinates equal in size still have to be read separately, one for each axis. Working: both coordinates of (−2, −2) are negative, so B lies to the left of the y-axis and below the x-axis; counting anticlockwise from the region where both coordinates are positive, left and below is the third region. Answer: the third quadrant. The distractors: 'the first quadrant' comes from ignoring both minus signs and treating the point as (2, 2); 'the second quadrant' comes from applying the minus sign to the x-coordinate only, as though the point were (−2, 2); 'the fourth quadrant' comes from applying it to the y-coordinate only, as though the point were (2, −2).
- (b) the first and the third — Method: the sign of a product depends only on the signs of the two numbers multiplied: like signs give a positive product and unlike signs a negative one, so look for the quadrants in which both coordinates carry the same sign. Working: in the first quadrant x and y are both positive, and positive times positive is positive; in the third quadrant both are negative, and negative times negative is positive as well; in the second quadrant x is negative while y is positive, and in the fourth x is positive while y is negative, so each of those gives a negative product. Answer: the first and the third. The distractors: 'the second and the fourth' comes from applying the rule for a NEGATIVE product, unlike signs, to a positive one; 'the first and the second' comes from testing only the y-coordinate and keeping the quadrants where the height is positive; 'the first and the fourth' comes from testing only the x-coordinate in the same way.
- (c) (2, 5) — Reflecting in the x-axis keeps the x-coordinate the same and changes the sign of the y-coordinate: Q = (2, 5). (−2, −5) comes from reflecting in the y-axis instead, which changes the sign of the x-coordinate. (−2, 5) comes from reflecting in both axes. (2, −5) comes from not applying the reflection at all.
- (d) 10 — The right angle is at (1, 1). The vertical side has length 5 − 1 = 4 and the horizontal side has length 6 − 1 = 5, so the area is (4 × 5) ÷ 2 = 20 ÷ 2 = 10. A candidate who forgets to halve the product of the two sides gets 4 × 5 = 20. A candidate who forgets to subtract the shared vertex's coordinate and uses the raw coordinates 6 and 5 as the side lengths gets (6 × 5) ÷ 2 = 30 ÷ 2 = 15. A candidate who uses only one side length as the area gets 5.
- (a) (−6, −3) — Reflecting in the y-axis changes the sign of the x-coordinate and keeps the y-coordinate the same: (−6, −3). (6, 3) comes from reflecting in the x-axis instead, which changes the sign of the y-coordinate. (−6, 3) comes from reflecting in both axes. (6, −3) comes from not applying the reflection at all.
- (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.
- (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.
- (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.
- (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.
- (d) the y-axis, and 12 — Method: coordinates are written (x, y), so the first number is the distance across and the second the distance up; a point whose first coordinate is 0 has not moved across from the origin and therefore lies on the vertical axis. Working: in (0, 12) the first number is 0, so P is on the y-axis, and the second number, 12, is the y-coordinate of P. Answer: the y-axis, and 12. The distractors: 'the x-axis, and 12' comes from mixing up which axis the condition 'the first coordinate is 0' describes; 'the y-axis, and 0' comes from placing P correctly but reading the pair the wrong way round, so that the first number is quoted as the y-coordinate; 'the x-axis, and 0' comes from making both of those mistakes at once.
- (b) (−4, 3) — Method: in a rectangle whose sides are parallel to the axes only two different x-coordinates and two different y-coordinates appear, and each of them is shared by a pair of vertices, so the missing vertex takes the x-coordinate and the y-coordinate that so far appear only once. Working: the x-coordinates given are −4, 2 and 2, so 2 is already used twice and −4 is used once; the y-coordinates given are −1, −1 and 3, so −1 is already used twice and 3 is used once; the fourth vertex therefore has x = −4 and y = 3. Answer: (−4, 3). The distractors: (3, −4) comes from picking the two unpaired coordinates correctly and then writing them in the wrong order; (−4, −5) comes from matching the 4-unit vertical side but measuring it downwards from (−4, −1) instead of upwards; (8, 3) comes from carrying on round the shape with the horizontal step used earlier, adding 6 to the x-coordinate of (2, 3) instead of closing the rectangle.
- (a) y = 0 — Method: in the pair (x, y) the first coordinate measures how far left or right of the origin a point is and the second how far above or below the x-axis it is, so a point on an axis has one of those measurements equal to zero. Working: the x-axis is the horizontal line through the origin, so a point sitting on it is neither above nor below that line and its second coordinate is zero, while its first coordinate may be positive, negative or zero. Answer: y = 0. The distractors: x = 0 is the condition for lying on the y-axis, the other axis; x > 0 comes from assuming a point on the x-axis must be to the right of the origin, which is true only of part of that axis; x = y holds only at the origin, which is one point of the x-axis rather than a property shared by all of them.
- (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.
Build your own mix at the worksheet builder.