Mathematics (Core)Exam code: 0580

Transformations

Transformations

PractiseFlashcards

Doing and describing one

  • A translation slides the shape without turning or resizing it, so the orientation is unchanged
  • The movement is given as a column vector, with the top number the horizontal move and the bottom number the vertical one
  • Positive top means right and negative means left; positive bottom means up and negative means down
  • Move every vertex by the same vector, then join them up
Shape P translated to its image P prime, with a red stepped path drawn from one vertex of P to the matching vertex of P prime, counting the horizontal steps first and then the vertical ones
Source: Translations by Save My Exams
  • To describe one, say translation and give the vector
  • Mark schemes also accept the movement in words, such as "2 right, 7 down"
Shape A translated to shape B, with a red stepped path drawn between one pair of matching vertices and annotated 2 right and 3 down, which is the column vector of the translation
Source: Translations by Save My Exams

Worked example

Translating by a column vector

Triangle A has vertices (5, 1), (9, 1) and (9, 3). Draw the image of A after a translation by the column vector with top number −4 and bottom number 5.

Solution:

  • The top number −4 means 4 squares to the left, and the bottom number 5 means 5 squares up
  • Move every vertex by the same amount: (5, 1) goes to (5 − 4, 1 + 5) = (1, 6)
  • In the same way, (9, 1) goes to (5, 6) and (9, 3) goes to (5, 8)
  • Join the three points with a ruler: the image has vertices (1, 6), (5, 6) and (5, 8)
  • Check: the image is the same size, shape and way round as A, and its base from (1, 6) to (5, 6) is horizontal and 4 long, like A's base from (5, 1) to (9, 1)
Triangle A with vertices (5, 1), (9, 1) and (9, 3) and its image A′ with vertices (1, 6), (5, 6) and (5, 8), with arrows from (5, 1) going 4 squares left and then 5 squares up to (1, 6)

Worked example

Describing a translation

Triangle A has vertices (−4, 2), (−2, 2) and (−2, 5). Triangle B has vertices (1, −1), (3, −1) and (3, 2). Describe fully the single transformation that maps A onto B.

A grid from x = −5 to 4 and y = −2 to 6 with triangle A, vertices (−4, 2), (−2, 2) and (−2, 5), and triangle B, vertices (1, −1), (3, −1) and (3, 2); no arrow or vector is drawn

Solution:

  • A and B are the same size, the same way up and not turned, so this is a translation
  • Take one vertex of A and its matching vertex on B: (−4, 2) goes to (1, −1)
  • Count from A to B: from −4 across to 1 is 5 to the right, and from 2 down to −1 is 3 down
  • So the column vector has top number 5 and bottom number −3
  • Check with a second pair: (−2, 5) goes to (3, 2), again 5 right and 3 down
  • The answer is a translation by the column vector with top number 5 and bottom number −3
  • Counting from B back to A gives the opposite signs, −5 and 3, so always count from the object to the image

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