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Cambridge IGCSE0580

Coordinates

Mathematics 0580 Chapter Notes

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Coordinates
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Coordinates notes

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In this chapter

  1. 1Reading and plotting a point
  2. 2Negative coordinates
  3. 3Finding a point from a vector

1. Reading and plotting a point

A point in two dimensions is fixed by two Cartesian coordinates, written (x, y). The x-axis runs across and the y-axis runs up.

The x-coordinate always comes first. Count along the x-axis, then up or down the y-axis. The axes cross at the origin, (0, 0).

(x, y) = (across, up or down)

Examiner insight

“The main error was to put the coordinates the wrong way round.” “Some candidates got the x and y coordinates confused and plotted at (2, –4) instead of (–4, 2).”

Worked example 12 marks

The point B is 4 units to the right of the y-axis and 2 units above the x-axis. Write down the coordinates of B.

  1. 1

    The x-coordinate is the distance across: x = 4 [1]

  2. 2

    The y-coordinate is the distance up: y = 2, so B is (4, 2) [1]

2. Negative coordinates

A point to the left of the y-axis has a negative x-coordinate. A point below the x-axis has a negative y-coordinate.

Keep the sign with its number. (-3, 2) and (3, -2) are different points.

Examiner insight

“A small number plotted the point (3, 3.11) just above y = 3 rather than on, or just above y = 3.1 and a few ignored the negative signs on the y values when plotting the points at x = –2 and x = –1.5.” “Candidates were more successful at plotting (–2, 4) although (–2, –4) was seen plotted a few times.”

Worked example 12 marks

The point C is 3 units to the left of the y-axis and 2 units below the x-axis. Write down the coordinates of C.

  1. 1

    Left of the y-axis, so x = -3 [1]

  2. 2

    Below the x-axis, so y = -2 and C is (-3, -2) [1]

3. Finding a point from a vector

The vector overrightarrowPQ = dbinomab is the movement from P to Q. It changes the x-coordinate by a and the y-coordinate by b.

The vector is not the coordinates of Q. Start from P, not from the origin, and add each component to the matching coordinate.

Take care with negative components. Adding a negative number moves the point left or down.

P(x, y) and overrightarrowPQ = dbinomab ⇒ Q = (x + a, y + b)

Examiner insight

“Candidates found plotting R challenging, with many taking the vector PR as the coordinates of R rather than the movement from P to R.” “Another common error was to start from P and move two squares and then one square but in the wrong directions; or doing the same thing but from the origin, not from P.”

Worked example 12 marks

P is the point (-3, 5) and overrightarrowPQ is dbinom8-3. Find the coordinates of the point Q.

  1. 1

    x-coordinate: -3 + 8 = 5 [1]

  2. 2

    y-coordinate: 5 + (-3) = 2, so Q is (5, 2) [1]

Recap

  • A point is written (x, y): x across first, then y up or down.
  • The axes meet at the origin, (0, 0).
  • Left of the y-axis is a negative x. Below the x-axis is a negative y.
  • The vector overrightarrowPQ = dbinomab takes P(x, y) to Q(x + a, y + b).
  • A vector is a movement from a point. It is not the coordinates of the new point.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write down the coordinates of the point 3 units to the right of the origin and 4 units up.1 mark
  2. A point A is 2 units to the left of the y-axis and 5 units above the x-axis. Write down its coordinates.2 marks
  3. Do not use a calculator. A point lies on the x-axis, 6 units to the left of the origin. Write down its coordinates.1 mark
  4. Do not use a calculator. ABCD is a rectangle with sides parallel to the axes. A is (1, 1), B is (5, 1) and C is (5, 4). Find the coordinates of D.2 marks
  5. The point P is (-4, 3). P moves 5 units to the right and 7 units down. Find its new coordinates.2 marks
  6. A is (-2, -3) and B is (4, -3). How many units is B to the right of A?2 marks
  7. P is the point (2, -7) and overrightarrowPQ is dbinom-59. Find the coordinates of Q.2 marks
  8. Do not use a calculator. Q is the point (6, -1) and overrightarrowPQ is dbinom4-5. Find the coordinates of P.2 marks
  9. Four posts A, B, C and D mark the corners of a field on a map grid, in that order. A is (-4, 1), B is (2, -3) and C is (5, 2). The movement from A to D is the same as the movement from B to C. Find the coordinates of D.3 marks
  10. Do not use a calculator. A drone starts at S(-2, 3) and flies to R. From R it flies by the vector dbinom4-5 and reaches T(1, -1). Find the coordinates of R and the vector overrightarrowSR.4 marks

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