Mathematics (Core)Exam code: 0580

Transformations

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
  • 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 a horizontal line

Triangle T has vertices (−1, 3), (3, 3) and (3, 5). Reflect T in the line y = 1, and give the vertices of its image.

Solution:

  • y = 1 is the horizontal line through every point whose y coordinate is 1, so draw it first, with a ruler, across the grid
  • A reflection in a horizontal line moves each point straight up or down, at right angles to the line, so every x coordinate stays the same
  • (−1, 3) is 2 squares above the line, so its image is 2 squares below it: (−1, −1)
  • (3, 3) is also 2 squares above, so its image is (3, −1)
  • (3, 5) is 4 squares above, so its image is 4 squares below: (3, −3)
  • Join the three points with a ruler: the image has vertices (−1, −1), (3, −1) and (3, −3)
  • Check: the line y = 1 is halfway between each point and its image, as 5 and −3 average to 1
  • Reflecting in x = 1 by mistake would move the points sideways instead, which is why the line goes in first
A grid from x = −3 to 6 and y = −4 to 6 with triangle T, vertices (−1, 3), (3, 3) and (3, 5), above the dashed mirror line y = 1, and its image T′, vertices (−1, −1), (3, −1) and (3, −3), below it; a dotted guide along x = 3 crosses the mirror at right angles

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

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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