Transformations flashcards
11 cards on Transformations: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 11
Transformations
Reflecting in a mirror line
How do you reflect a shape in a mirror line such as x = 2 or y = −1?
Answer it in your head, then turn the card over.
Reflecting in a mirror line
Key points
- Each image point is the same distance from the mirror line as the original, on the other side.
- Measure at right angles to the mirror line.
- x = a is a vertical line; y = b is a horizontal line.
- Reflect each corner, then join them up. Tracing paper is allowed.
Reflect (5, 1) in x = 2
(5, 1) is 5 − 2 = 3 squares right of the line.
Its image is 3 squares left of the line: 2 − 3 = −1, so (−1, 1)
Watch out
Every point on x = 2 has an x-coordinate of 2, so x = 2 runs up and down; y = 2 runs across.
Transformations
Reflecting in y = x and the axes
What happens to a point's coordinates when it is reflected in y = x, y = −x or one of the axes?
Answer it in your head, then turn the card over.
Reflecting in y = x and the axes
Key points
- In y = x the coordinates swap: (a, b) goes to (b, a).
- In y = −x they swap and both change sign: (a, b) goes to (−b, −a).
- In the x-axis the y-coordinate changes sign: (a, b) goes to (a, −b).
- In the y-axis the x-coordinate changes sign: (a, b) goes to (−a, b).
Watch out
(3, 1) reflected in y = x is (1, 3); (3, −1) is its reflection in the x-axis, a different line.
Transformations
Rotating a shape
What do you need to rotate a shape, and how do you carry out the rotation?
Answer it in your head, then turn the card over.
Rotating a shape
Key points
- A rotation needs an angle, a direction (clockwise or anticlockwise) and a centre.
- Trace the shape, hold the tracing at the centre with your pencil point, and turn it.
- A half turn (180°) looks the same either way, so it needs no direction.
- Turning one way is the same as turning 360° minus the angle the other way.
270° clockwise
360 − 270 = 90
So 270° clockwise gives the same image as 90° anticlockwise.
Watch out
Check which way you turn: 90° clockwise and 90° anticlockwise land the shape in different places.
Transformations
Finding a centre of rotation
Given a shape and its image after a rotation, how do you find the centre of rotation?
Answer it in your head, then turn the card over.
Finding a centre of rotation
Key points
- The centre is the one point that does not move.
- Try points with tracing paper until the turn lands the shape exactly on its image.
- The centre is the same distance from each corner as from that corner's image.
- For a half turn only, it is the midpoint of any point and its image.
A half turn takes (1, 2) to (5, 4)
Midpoint: ((1 + 5) ÷ 2, (2 + 4) ÷ 2)
The centre is (3, 3).
Watch out
The midpoint trick only works for 180°: for a quarter turn, find the point that stays still.
Transformations
Translating with a column vector
How does a column vector describe a translation, and how do you carry one out?
Answer it in your head, then turn the card over.
Translating with a column vector
Key points
- A translation slides every point the same distance in the same direction: no turning, flipping or resizing.
- In the column vector, the top number moves across (positive is right), the bottom up or down (positive is up).
- Move each corner by the vector, then join the corners up.
Translate (2, 5) by (−3 over 2)
3 left and 2 up: (2 − 3, 5 + 2)
The image is (−1, 7).
Watch out
The top number is always the move across: (−3 over 2) is 3 left and 2 up, not 3 down and 2 right.
Transformations
Enlarging from a centre
How do you enlarge a shape by a scale factor from a centre of enlargement?
Answer it in your head, then turn the card over.
Enlarging from a centre
Key points
- Scale factor = new length ÷ old length.
- Count across and up from the centre to each corner.
- Multiply both distances by the scale factor, plot the new corner, then join them up.
- Each new corner lies on the line from the centre through its old corner.
Centre (1, 1), corner (3, 2), factor 3
From the centre: 2 right and 1 up
× 3: 6 right and 3 up
New corner: (1 + 6, 1 + 3) = (7, 4)
Watch out
Multiply the distances from the centre: adding 3 to them changes the shape, and the right size in the wrong place loses marks.
Transformations
Fractional scale factors
What does a scale factor between 0 and 1 do, and why is it still called an enlargement?
Answer it in your head, then turn the card over.
Fractional scale factors
Key points
- It makes the image smaller, closer to the centre.
- An enlargement is any change of size from a centre, so it is still called one.
- Every distance from the centre, and every length, is multiplied by the fraction.
- The angles stay the same, so the image is similar to the original.
Sides go from 6 cm to 3 cm
Scale factor = 3 ÷ 6 =
A corner 4 right and 6 up from the centre moves to 2 right and 3 up.
Watch out
A smaller image means a fraction: sides from 6 cm to 3 cm give , not 2.
Transformations
HigherNegative scale factors
What does a negative scale factor do to an enlargement?
Answer it in your head, then turn the card over.
Negative scale factors
Key points
- The image goes on the opposite side of the centre, and is upside down.
- The size of the number still sets the distance: −2 doubles every distance from the centre.
- Multiply both distances from the centre by the scale factor, so their signs flip.
- Scale factor −1 gives the same image as a half turn about the centre.
Centre (2, 1), corner (3, 3), factor −2
From the centre: 1 right and 2 up
× −2: 2 left and 4 down
New corner: (2 − 2, 1 − 4) = (0, −3)
Watch out
A negative scale factor never keeps the image on the same side, and it does not make the image smaller.
Transformations
Describing a transformation fully
What details must you give to describe each type of transformation fully?
Answer it in your head, then turn the card over.
Describing a transformation fully
Key points
- Reflection: the equation of the mirror line, such as y = x.
- Rotation: the angle, the direction and the centre.
- Translation: the column vector.
- Enlargement: the scale factor and the centre.
- Always one single transformation, named, with every detail.
Watch out
"Rotation 90° about (1, 2)" is not full: it needs clockwise or anticlockwise as well.
Transformations
Spotting which transformation it is
Looking at a shape and its image, how do you tell which transformation it is?
Answer it in your head, then turn the card over.
Spotting which transformation it is
Key points
- Same size and facing the same way: a translation. Every side stays parallel to its original.
- Same size and flipped, a mirror image: a reflection.
- Same size and turned: a rotation.
- Different size, bigger or smaller: an enlargement.
Watch out
"A reflection and then a translation" scores nothing: the question wants one transformation that does the job.
Transformations
Finding a centre of enlargement
Given a shape and its enlargement, how do you find the scale factor and the centre?
Answer it in your head, then turn the card over.
Finding a centre of enlargement
Key points
- Scale factor = a length on the image ÷ the matching length on the original.
- Join each corner of the image to its matching corner on the original, and extend the lines.
- The lines meet at the centre of enlargement.
- Check with one more pair of corners: their line must pass through the centre too.
A side of 2 cm becomes 6 cm
Scale factor = 6 ÷ 2 = 3
Watch out
Divide image by original, not the other way: 2 ÷ 6 would give for an image that got bigger.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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