MathematicsExam code: 0580

Transformations

Vectors and Transformations

PractiseFlashcards

Doing one

  • A reflection flips the shape across a mirror line, so the image is the same size but the wrong way round
  • Every point and its image sit the same perpendicular distance from the mirror line, on opposite sides
A square object ABCD and its image across a dashed vertical mirror line, each vertex marked with a cross: A sits nearest the line and its image A prime sits the same number of squares on the other side, and the labelling order runs the opposite way round
Source: Reflections by Save My Exams
  • Take one vertex at a time, count squares straight across to the line at right angles, then continue the same count past it
A shape reflected in the dashed line x = −1, with dashed guide lines drawn from two of its vertices straight across to the mirror line and the same distance again beyond it, each meeting the line at a marked right angle
Source: Reflections by Save My Exams
  • For a diagonal mirror line such as y = x, count diagonally across the squares rather than horizontally
  • Any vertex sitting on the line itself stays exactly where it is, making it invariant
  • Where the mirror line passes through the shape, part of it reflects each way

Worked example

Reflecting in the line y = x

Triangle A has vertices (−1, 6), (3, 6) and (3, 8). Reflect A in the line y = x and draw its image.

Solution:

  • Draw the mirror line first, with a ruler: y = x runs diagonally through (0, 0), (1, 1), (2, 2) and so on
  • A reflection in y = x swaps the two coordinates of every point, so (a, b) goes to (b, a)
  • (−1, 6) goes to (6, −1), (3, 6) goes to (6, 3), and (3, 8) goes to (8, 3)
  • Check one pair: (−1, 6) and (6, −1) have midpoint (2.5, 2.5), which lies on y = x, so they sit the same distance either side of the line
  • Plot the three points and join them with a ruler: the image has vertices (6, −1), (6, 3) and (8, 3)
  • Swapping the coordinates and also changing both signs would reflect in y = −x instead, sending (−1, 6) to (−6, 1)
A grid with the dashed mirror line y = x. Triangle A has vertices (−1, 6), (3, 6) and (3, 8), and its reflection A′ has vertices (6, −1), (6, 3) and (8, 3), on the other side of the line; a dotted guide joins (−1, 6) to (6, −1), crossing the mirror at right angles at (2.5, 2.5)

Describing one

  • Say reflection, then state which line the shape was flipped across, written as an equation
  • Vertical lines are x = k and horizontal lines are y = k, which is the pair most often swapped over
  • The diagonal through the origin sloping up is y = x, and the one sloping down is y = −x

Worked example

Describing a reflection fully

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

A coordinate grid with x from −4 to 4 and y from 0 to 6. Triangle A is drawn with vertices at (1, 2), (3, 2) and (1, 5), and triangle B with vertices at (−1, 2), (−3, 2) and (−1, 5). No mirror line is drawn.

Solution:

  • The shape is the same size and the same way up, but it has been flipped, so this is a reflection
  • Compare one pair of matching points: (1, 2) becomes (−1, 2)
  • The y coordinate is unchanged and the x coordinate has changed sign
  • The mirror line is therefore vertical, midway between 1 and −1, which is x = 0
  • Check on a second pair: (3, 2) becomes (−3, 2), midway again at x = 0
  • The answer is a reflection in the line x = 0
  • Give the mirror line as its equation: x = 0 is the y-axis

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