Printable · GCSE Higher · ages 14-16
Combinations of transformations and invariance worksheet — GCSE Higher
Fifteen questions on "combinations of transformations and invariance" — DfE statement G8. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Combinations of transformations and invariance worksheet — GCSE Higher
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- 1.A game designer places a coin at (5, 3) on a grid. The game rotates the coin 90° clockwise about the point (2, 3) each time the player presses a button, and this is applied twice in a row. Work out the coordinates of the coin after the button is pressed twice.
- 2.A shape is reflected in the line x = 1, and the image is then reflected in the line x = 5. Which single transformation is equivalent to this combination, for every point?
- 3.A rotation of 200° clockwise about the origin is applied, and the image is then rotated 250° anticlockwise about the origin. Work out the single angle of rotation, measured clockwise and between 0° and 360°, that has the same overall effect as the two rotations combined.
- 4.An engineer models a technical drawing on a coordinate grid. A component is reflected in the line x = 2, and the image is then reflected in the line x = 6. Which single transformation has the same overall effect as these two reflections, for every point on the drawing?
- 5.A line passes through the points (1, 2) and (3, 6). The line is reflected in the x-axis, and the image is then translated by the vector (2, 0). Work out the gradient of the image line after both transformations.
- 6.A trapezium has vertices (0, 2), (4, 2), (3, 5) and (1, 5). It is rotated 180° about the point (2, 2), and the image is then translated by the vector (1, −5). Work out the coordinates of the image of (0, 2).
- 7.Line l1 has equation x = 2, and line l2 has equation y = x − 1. They intersect at a single point. A shape is reflected in l1, and the image is then reflected in l2. Which point is invariant under this combined transformation?y = x − 1
- 8.A shape is reflected in the line y = x, and the image is then reflected in the line y = 0. These two lines meet at the origin. Work out the single rotation, centre and angle, that is equivalent to this combination of two reflections, for every point.y = x
- 9.Point Q has coordinates (3, 5). It is translated by the vector (1, −2), and the image is then enlarged by scale factor 3, centre the origin. Work out the coordinates of the final image of Q.
- 10.A shape is translated by the vector (5, −3), and the image is then translated by the vector (−5, 3). Which single transformation has the same overall effect as this combination, for every point?
- 11.A rotation of 180° about the point (3, 2) can be written as a reflection in the line x = 3, followed by a reflection in a second line, for every point in the plane. Which line is the second line?
- 12.Square B has vertices (1, 1), (4, 1), (4, 4) and (1, 4). It is rotated 90° clockwise about the origin, and the image is then translated by the vector (2, −1). Work out the coordinates of the image of (4, 1).
- 13.Point P is translated by the vector (5, −2), and the image is then rotated 90° clockwise about the origin to give the point (3, 9). Work out the coordinates of P.
- 14.Triangle S has vertices (2, 2), (5, 2) and (2, 5). It is mapped onto triangle S′ with vertices (2, 5), (5, 5) and (2, 2). Which single composition of two transformations maps S onto S′?
- 15.Triangle T has an area of 6 cm². It is enlarged by scale factor −2 about the origin, and the image is then rotated 90° about the origin. Work out the area of the final image, and state whether it is the same way round as T or a mirror image of it.
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