0580

Transformations

Vectors and Transformations

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

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
Shape A to the left of the axis and its image B to the right, reflected in the dashed red vertical line x = −1, with matching vertices the same distance either side of it
Source: Reflections by Save My Exams
Exam tip

Give the mirror as an equation. Nine of the 43 mark schemes award a mark for naming the right kind of line — x = k or y = k — even when the number itself is wrong, so a wrong equation still beats no equation. "In the y-axis" on its own does not: write x = 0.

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
  • Both parts are needed: "reflection" alone scores 1 of the 2, and the mirror must be given as an equation, not as "the y-axis" alone