How do I enlarge a shape from a centre that is not the origin?

Find the vector from the centre to each vertex, multiply it by the scale factor, and add it to the centre. You can also draw rays from the centre.

What does a negative scale factor do?

The image is on the opposite side of the centre and upside down. The size is multiplied by the number ignoring its sign. This is Higher tier.

Does the area scale by the scale factor?

No. Lengths scale by k and areas scale by k squared. A scale factor of 3 makes the area 9 times bigger.

Year 11 transformations revision notes to download as PDF

Q: What do I write to describe a rotation?

Give three things: the angle, the direction (clockwise or anticlockwise) and the centre of rotation. Missing one loses a mark.

This Year 11 transformations revision notes is written for pupils aged 15-16 and follows the National Curriculum. In short, it shows you how to describe and perform transformations on grids. Before you start, it helps to revise iteration, growth and decay and constructions, loci, bearings, plans and elevations, because this lesson builds on them.

Key points

A reflection is described by its mirror line, for example \(x=5\) or \(y=x\); each point and its image are the same distance from the mirror.
A rotation is described by its angle, direction and centre; about the origin \(90^\circ\) anticlockwise sends \((x,y)\) to \((-y,x)\) and \(180^\circ\) sends \((x,y)\) to \((-x,-y)\).
To rotate about another centre, rotate the vector from the centre to the point, then add the centre back.
A translation is described by a column vector \(\begin{pmatrix}a\\b\end{pmatrix}\) and moves every point by the same amount; the image vector is image minus object.
An enlargement needs a scale factor and a centre; each point is moved to \(c+k(p-c)\), and a scale factor between 0 and 1 gives a smaller image.
(Higher) A negative scale factor puts the image on the opposite side of the centre; for \(k=-2\) and centre \((1,1)\), the point \((3,2)\) goes to \((-3,-1)\).

Table of Contents

Year 11 transformations revision notes: what you will learn

At GCSE there are four transformations: reflection, rotation, translation and enlargement. Negative scale factors and combinations of transformations are Higher tier only (marked Higher).

Starter activity

Draw an arrow shaped like the letter F on squared paper. Use tracing paper to show what the shape looks like after a reflection in a vertical line, after a turn of a quarter-turn about a corner and after a slide to the right. Which of these moves changes the way the F faces, and which keeps it the same? Explain how you can tell by looking at the letter.

Reflection

Definition: transformation, object and image

A transformation moves or changes a shape. The original shape is the object and the new shape is the image.

Rule: reflection

To describe a reflection you must give the mirror line. Each point and its image are the same distance from the mirror, on opposite sides, and the line joining them is perpendicular to the mirror. Reflecting \((x,y)\) in the \(x\)-axis gives \((x,-y)\); in the \(y\)-axis gives \((-x,y)\); in \(y=x\) gives \((y,x)\); in \(y=-x\) gives \((-y,-x)\). For a mirror like \(x=4\), count the distance: the reflection of \((2,5)\) is \((6,5)\) because the point is 2 to the left of the line.

Rotation

Rule: rotation

To describe a rotation you must give three things: the angle, the direction (clockwise or anticlockwise) and the centre. About the origin:

\(90^\circ\) clockwise: \((x,y)\to(y,-x)\)
\(90^\circ\) anticlockwise: \((x,y)\to(-y,x)\)
\(180^\circ\): \((x,y)\to(-x,-y)\)

Method: rotating about another centre

Subtract the centre from the point.
Rotate using the rule for the origin.
Add the centre back. Tracing paper is allowed: trace the shape, put your pencil on the centre and turn the paper.

Worked example 1: rotation about another centre

Rotate \((4,2)\) by \(90^\circ\) clockwise about \((1,1)\).

Relative to the centre the point is \((3,1)\). Rotating gives \((1,-3)\). Adding the centre back gives \((2,-2)\).

Translation

Definition: translation

A translation slides every point by the same amount, given as a column vector \(\begin{pmatrix}a\\b\end{pmatrix}\). The top number is the move right (negative means left) and the bottom number is the move up (negative means down). The shape does not turn or change size. To describe a translation, give only the vector.

Enlargement

Definition: enlargement

To describe an enlargement give the scale factor and the centre. Every length is multiplied by the scale factor, but angles stay the same, so the image is similar to the object. A scale factor between 0 and 1 makes the shape smaller.

Worked example 2: enlarging a point

Enlarge \(A(3,2)\) by scale factor 2 about \((1,1)\).

The vector from the centre is \((2,1)\), doubled to \((4,2)\), so the image is \((5,3)\).

Rule: scale factor and area

Lengths scale by \(k\), but areas scale by \(k^2\). A rectangle \(2\ \text{cm}\times3\ \text{cm}\) enlarged by scale factor 4 is \(8\times12\), so the area is 96 cm\(^2\), which is \(6\times16\).

(Higher) Negative scale factors and combinations

Key fact (Higher): negative scale factors

With a negative scale factor the image is on the opposite side of the centre and appears upside down. The size is multiplied by the size of the scale factor, ignoring the sign. Enlarging \((4,2)\) with scale factor \(-2\), centre the origin, gives \((-8,-4)\). With scale factor \(-1\) the shape is turned through \(180^\circ\) about the centre.

Key fact (Higher): combinations

Do the transformations in the order written, applying the first one to the original shape and the second to the new image. Some combinations equal a single transformation:

Two reflections in parallel lines make a translation, twice as far as the distance between the mirrors and perpendicular to them.
Two reflections in lines that meet make a rotation about the meeting point.
Reflecting in the \(x\)-axis then the \(y\)-axis (or the other way round) is a rotation of \(180^\circ\) about the origin.

Describing transformations

Worked example 3: describe fully

Triangle \(T\) has vertices \((2,1),(4,1),(2,2)\). Its image has vertices \((-1,2),(-1,4),(-2,2)\). Describe the transformation.

Test the rule \((x,y)\to(-y,x)\): \((2,1)\to(-1,2)\) and \((4,1)\to(-1,4)\). It works. The transformation is a rotation of \(90^\circ\) anticlockwise about \((0,0)\).

Worked example 4 (Higher): one transformation or two?

Point \(P(2,3)\) is reflected in the \(x\)-axis and then in the \(y\)-axis. Describe the single transformation.

The point goes to \((2,-3)\) and then to \((-2,-3)\). This is a single rotation of \(180^\circ\) about the origin.

Note: what you must state

Transformation	You must state
Reflection	the mirror line, such as \(y=x\) or \(x=3\)
Rotation	angle, direction, centre
Translation	the column vector
Enlargement	scale factor and centre

Only give one transformation when the question says "single transformation". Mixing words like "move and turn" gains no marks.

Watch out: common mistakes

Leaving out the centre of a rotation or enlargement.
Writing "move" instead of "translation".
Confusing the vector signs: \(\begin{pmatrix}-3\\2\end{pmatrix}\) is 3 left and 2 up.
Using the scale factor for the area instead of its square.

Worked example 5: Describing a reflection

Triangle \(A\) has a vertex at \((2,5)\) and its image has a vertex at \((8,5)\). Describe the reflection.

The \(y\)-coordinates are equal, so the mirror line is vertical. The mirror is halfway between the two \(x\)-values: \((2+8)\div2=5\). The reflection is in the line \(x=5\). For a reflection always give the equation of the mirror line, and check using a second point.

Worked example 6: Finding the centre of a rotation

A shape rotates \(90^\circ\) clockwise and the point \((5,1)\) goes to \((3,-3)\). Find the centre of rotation.

Let the centre be \((a,b)\). The vector from the centre to the point is \((5-a,1-b)\). A clockwise quarter turn sends \((x,y)\) to \((y,-x)\), giving \((1-b,\ a-5)\). Adding the centre, the image is \((a+1-b,\ a+b-5)=(3,-3)\). So \(a-b=2\) and \(a+b=2\), which gives \(a=2\) and \(b=0\). The centre is \((2,0)\). Check: the vector \((3,1)\) turns into \((1,-3)\) and \((2,0)+(1,-3)=(3,-3)\) as required.

Worked example 7: Using vectors for a translation

The point \((7,-2)\) is the image of \((3,4)\) under a translation. Find the translation vector.

The vector is image minus object: \(\begin{pmatrix}7-3\\-2-4\end{pmatrix}=\begin{pmatrix}4\\-6\end{pmatrix}\). In words, 4 units to the right and 6 units down. A translation moves every point of the shape by the same vector, so the image is congruent to the object.

Worked example 8: Enlargement with a fractional scale factor

Enlarge the triangle \((2,2)\), \((6,2)\), \((2,6)\) by scale factor \(\tfrac12\) with centre \((0,0)\).

Multiply every coordinate by \(\tfrac12\): \((1,1)\), \((3,1)\), \((1,3)\). A scale factor between 0 and 1 makes the shape smaller (a reduction) but still an enlargement in the mathematical sense. The image lies on the same side of the centre as the object.

Worked example 9: Identifying the centre from corresponding points

Under an enlargement with scale factor 3, \((2,3)\) goes to \((4,9)\). Find the centre.

Let the centre be \((a,b)\). Then \(a+3(2-a)=4\), so \(6-2a=4\) and \(a=1\). And \(b+3(3-b)=9\), so \(9-2b=9\) and \(b=0\). The centre is \((1,0)\). Check: \((1,0)+3(1,3)=(4,9)\) ✓.

Worked example 10: The effect of transformations on lengths and angles

Which transformations make a congruent image and which make a similar image?

Reflections, rotations and translations keep all lengths and angles the same, so the image is congruent. An enlargement keeps angles but changes lengths by the scale factor, so the image is similar. Reflections reverse the orientation (clockwise labels become anticlockwise), while rotations and translations keep it.

Worked example 11: A transformation described by coordinates

Describe the transformation \((x,y)\to(y,x)\) and the transformation \((x,y)\to(-y,x)\).

The first swaps the coordinates: it is a reflection in the line \(y=x\). The second, \((1,0)\to(0,1)\) and \((0,1)\to(-1,0)\), is a rotation of \(90^\circ\) anticlockwise about the origin. Testing one or two points is the quickest way to identify a transformation.

Note: what you must state

To earn full marks when you describe a transformation, give all the details. For a reflection give the mirror line. For a rotation give the angle, the direction (clockwise or anticlockwise) and the centre. For a translation give the column vector. For an enlargement give the scale factor and the centre. Only one transformation should be described when the question says "single". Use tracing paper to check the centre of a rotation.

Method: rotating about a point that is not the origin

Find the vector from the centre to the point: point minus centre.
Rotate this vector: \(90^\circ\) anticlockwise maps \((x,y)\to(-y,x)\), \(90^\circ\) clockwise maps \((x,y)\to(y,-x)\), and \(180^\circ\) maps \((x,y)\to(-x,-y)\).
Add the rotated vector to the centre to get the image.
Check that the image is the same distance from the centre as the original.

Watch out: errors with transformations

A reflection needs a mirror line equation, not just the word "flip".
Do not say "rotation" without the centre, angle and direction.
For an enlargement, multiply the distances from the centre, not from the origin, unless the centre is the origin.
Area scale factors are the square of the length scale factor, even for a negative scale factor.

Did you know?

Patterns made by reflections, rotations and translations decorate buildings all over the world. The Alhambra palace in Granada, Spain, which was built about 700 years ago, contains tilings that use all these transformations, and the Dutch artist M. C. Escher studied them there before making his famous tessellations of birds and fish. Today computer games use the same mathematics to rotate, reflect and enlarge images on a screen many times each second.

Otto’s tip

Check your description by testing one vertex. If your rule sends one corner to the right place and another corner to the right place, it is almost certainly correct.

How to use this Year 11 transformations revision notes

First, read the key points so that you know where the Year 11 transformations revision notes is going. Then work through each definition and worked example, and cover the solution before you read it. After that, practise with the transformations worksheet and check every question with the worksheet answers. When you feel ready, test yourself with the multiple choice questions or the quick quiz. Finally, sit the transformations test and mark it with the test answers.

Also, the starters are a good warm-up, so you can keep the key facts fresh. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 11 transformations revision notes gives you everything you need for the chapter.

To sum up, regular practice works better than one long session. For instance, ten minutes a day is often enough. Besides, mistakes are useful, because they show what to revise next. Then, once a method feels easy, try a harder question. As a result, you build confidence step by step.