Paper Together: sit a paper in a room with your friends.See how it works
Transformations: worked examples and method
Foundation and Higher · Grade 3 · Non-calculator
This page sets out the method for transformations in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Transformations topic page.
Key facts
Translation: column vector (ab) means a right (left if negative) and b up (down if negative). Every point moves the same way.
Reflection: describe with the equation of the mirror line. Each point ends up the same distance the other side.
Rotation: describe with angle, direction and centre. 180° needs no direction.
Enlargement: describe with scale factor and centre. Distances from the centre are multiplied by the scale factor.
Translations, reflections and rotations keep the shape congruent; enlargement makes it similar (same shape, different size).
Rotating 90° clockwise about the origin sends (x,y) to (y,−x); 180° sends (x,y) to (−x,−y).
The method, step by step
Method: describe fully
When you see 'describe fully the single transformation', name the type and give every detail it needs.
Decide the type: same size and same way up (translation), flipped (reflection), turned (rotation), different size (enlargement).
Translation: write the vector. Reflection: write the line's equation.
Rotation: give the angle, the direction and the centre as a coordinate.
Enlargement: give the scale factor and the centre.
Only ever describe one transformation — 'a reflection then a translation' scores nothing.
Method: rotate with tracing paper
When you must rotate a shape, use tracing paper — it is allowed and saves marks.
Trace the shape and mark the centre of rotation.
Hold your pencil on the centre and turn the paper by the angle in the given direction.
Press the corners through and join them.
Check each vertex is the same distance from the centre as before.
Method: enlarge from a centre
When you see 'enlarge by scale factor k, centre (a,b)', work with distances from the centre.
For each vertex, count how far it is from the centre (across and up).
Multiply both distances by the scale factor.
Count that new distance from the centre and plot the new vertex.
Join the vertices and check the side lengths are k times bigger.
How the lesson explains it
Translation
Every point moves by the same vector.
The vector (3, −1) means 3 right and 1 down. The shape does not turn.
Reflection
Every point is the same distance the other side of the mirror line.
Name the line. “The y-axis” or “x = 1” is a full description.
Rotation
A rotation needs a centre, an angle and a direction.
90° clockwise about (0, 0) sends (1, 0) to (0, −1).
Enlargement
An enlargement needs a centre and a scale factor.
Scale factor 2 doubles every distance from the centre. Scale factor 1 leaves the shape where it is.
Describe fully
A question that says “describe the transformation” wants the type and the extra facts.
“It moved” is not a description.
Congruent or similar
Translation, reflection and rotation keep the size. The image is congruent.
An enlargement changes the size. The image is similar.
Enlargement and a full description
An enlargement changes size from a centre. Scale factor 2 doubles the distance from the centre to each vertex. It does not double the coordinates unless the centre is (0,0). For centre (1,1) and point C(4,1), the vector from the centre is (4-1, 1-1)=(3,0). Double it to get (6,0). Add that to the centre: (1+6, 1+0)=(7,1). Point B(1,3) has vector (0,2) from the centre, which doubles to (0,4), so the image is (1,5). Point A(1,1) is the centre, so it stays at (1,1).
A complete enlargement description needs the scale factor and the centre. Scale factor 1 leaves the shape as it is. A scale factor between 0 and 1, such as 1/2, makes a smaller image, still called an enlargement in this topic. If two shapes are similar and you are asked to describe the single transformation, check the size first. Same size means translation, reflection or rotation. Different size means enlargement, and you still have to find the centre by drawing rays through matching vertices until they meet.
Method: 1. Count the vector from the centre to the point, multiply by the scale factor, then add the centre back. 2. From centre (1,1) to (4,1), the vector (3,0) doubles to (6,0), so the image is (7,1). 3. Name both the scale factor and the centre in the description.
What "describe fully" needs
When a question says "describe fully the single transformation", the marks are for the type and for each detail. For a translation you need the type and the column vector. For a reflection you need the type and the equation of the mirror line, such as x = 3 or y = x. For a rotation you need the angle, the direction (clockwise or anticlockwise) and the centre. For an enlargement you need the scale factor and the centre.
A single transformation is never "a reflection and a translation". If two types are needed, the question would say so. Work out which type it is by comparing the shapes. If the shape faces a different way and is the same size, it is a rotation or a reflection. If the corners are in a mirror order, it is a reflection. If the size changes, it is an enlargement. If the shape slides without turning, it is a translation.
Method: 1. Compare the two shapes: same size or different, same way round or turned or flipped. 2. Name the type of transformation in one word. 3. Find the details the type needs: vector, mirror line, angle with direction and centre, or scale factor with centre. 4. Check the description by moving one vertex with your rule and seeing that it lands where it should.
Tip: Use one vertex as a test. If your description sends that corner to the right place and the shape is not stretched, the whole description is almost certainly right.
Reflection in lines such as x = 3 and y = x
To reflect a point in a vertical line x = a, keep the y coordinate and find the new x by measuring the same distance on the other side of the line. The rule is x to 2a - x. For example, reflecting (5, 2) in x = 3 gives (1, 2), because 5 is 2 to the right of 3 and 1 is 2 to the left. For a horizontal line y = b, keep x and use y to 2b - y.
The two diagonal lines have simple rules. Reflection in y = x swaps the coordinates, so (a, b) to (b, a). Reflection in y = -x swaps them and changes both signs, so (a, b) to (-b, -a). Exam questions on these lines often start from a point on a grid, so if you are unsure of the rule, test it with a simple point such as (2, 1) on a sketch.
Method: 1. Draw the mirror line on a quick sketch and plot the point. 2. Count the distance from the point to the line, then count the same distance on the other side. 3. Use the rules x to 2a - x for x = a, and y to 2b - y for y = b. 4. Check that the point and its image are the same distance from the line.
Enlargement with a centre, negative and fractional factors
To enlarge a shape from a centre, take the vector from the centre to each vertex and multiply it by the scale factor. If the centre is (1, 1) and a vertex is (4, 1), the vector is (3, 0). With scale factor 2 the new vector is (6, 0), so the image vertex is (7, 1). With a fractional factor such as 1/2 the shape gets smaller. With a negative factor the image is on the opposite side of the centre and is upside down.
Lengths are multiplied by the scale factor k and areas by k^2. A scale factor of -2 makes lengths twice as long and areas four times as big, just as 2 does. Angles never change. In exam questions that use the phrase "what is the ratio of the areas?", the answer is 1 : k^2, and you should remember that a negative k still gives a positive k^2.
Method: 1. Find the vector from the centre to each vertex. 2. Multiply each vector by the scale factor, keeping the signs. 3. Add the new vector to the centre to get the image vertex. 4. For areas, multiply by k^2. For lengths, multiply by |k|.
Tip: Check a long side. If the original side was 3 and the scale factor is -2, the image side must be 6 long.
Doing two transformations in a row
Some questions ask for the image after one transformation followed by another, or for a single transformation that does the work of two. Do them one at a time and write the coordinates after each step. The order can matter: a reflection followed by a translation is usually different from the translation followed by the reflection.
To find the single equivalent transformation, track one or two points through both steps and then look at where they finish. If the points move by the same vector, it is a translation. If one coordinate changes sign and the other does not, it is a reflection in an axis. A good check is the invariant point, which is a point that does not move. A point on a mirror line and the centre of a rotation are both invariant.
Method: 1. Apply the first transformation to a point and write down its image. 2. Apply the second transformation to that image. 3. Compare the start and the finish to identify the single transformation. 4. Test a second point to be sure the rule works.