Advanced problem solving with further transformations | Maths

Starter quiz

A point on a shape is ______ if that point has not changed location after the shape is transformed.

'invariant' ✓
Select the statements which fully describe a transformation.

enlargement by scale factor 3

reflection in the $y$ -axis ✓

rotation of 90° about the origin
What type of transformation maps shape B onto shape A?

A reflection in the line $x=0$

A reflection in the line $y=0$ ✓

A rotation of 180° about (3, 0) ✓

A rotation of 90° about (0 0)

An enlargement scale factor -1 about (3, 0) ✓
Describe one possible transformation from A to B that has a collection of invariant points.

An enlargement scale factor -1 about $(3, 2)$

A reflection in the line $x=3$ ✓

A rotation of $180$ ° about $(3, 2)$

A translation by the vector $\begin{pmatrix} 2 \\ 0 \\ \end{pmatrix}$
Select the two transformations needed to map shape H onto shape E.

An enlargement scale factor $\frac{1}{2}$ about centre $(-5, 4)$

An enlargement scale factor $\frac{1}{2}$ about centre $(-6, 5)$ ✓

followed by a translation by $\begin{pmatrix} 8 \\ -5 \\ \end{pmatrix}$ ✓

followed by a translation by $\begin{pmatrix} 8 \\ 5 \\ \end{pmatrix}$
Select the two transformations needed to map shape B onto shape G.

A reflection in the line $y$ = 0 ✓

A reflection in the line $x$ = 0

followed by a reflection in the line $y=x$

followed by a rotation of 90° clockwise about the origin ✓

followed by a rotation of 90° clockwise about the (0, -1)

Exit quiz

The object is the starting figure, before a transformation has been applied. The ______ is the resulting figure, after a transformation has been applied.

'image' ✓
Shape A is mapped onto shape B by a reflection. The equation of line that contains the invariant points is ______.

'x=3' ✓
Sofia transforms an object by the vector $\begin{pmatrix} -4 \\ 3 \\ \end{pmatrix}$ onto its image. What transformation will map the image back onto the object?

A translation by $\begin{pmatrix} -4 \\ -3 \\ \end{pmatrix}$

A translation by $\begin{pmatrix} 4 \\ -3 \\ \end{pmatrix}$ ✓

A translation by $\begin{pmatrix} -3 \\ 4 \\ \end{pmatrix}$

A translation by $\begin{pmatrix} 4 \\ 3 \\ \end{pmatrix}$

A translation by $\begin{pmatrix} 3 \\ -4 \\ \end{pmatrix}$
Jacob enlarges an object by a scale factor of $-2$ about $(2, -3)$ . What transformation will map the image back onto the object?

Enlargement scale factor $2$ about $(-2, 3)$

Enlargement scale factor $\frac{1}{2}$ about $(2, 3)$

Enlargement scale factor $-\frac{1}{2}$ about $(-2, 3)$

Enlargement scale factor $-2$ about $(2, -3)$

Enlargement scale factor $-\frac{1}{2}$ about $(2, -3)$ ✓
Some Oak pupils are discussing invariant points after a transformation. Which pupils are correct?

Jun: A rotation carried out about any point P on an object has P as invariant ✓

Sam: A rotation of a triangle can give a single line of invariant points

Andeep: A translation never gives an invariant point unless the vector is 0 ✓

Aisha: A reflection about a line on the shape gives a line of invarient points ✓

Jacob: An enlargement about a point not on the object gives one invariant point
$Shape I is enlarged by s.f. <Math>\frac{1}{2}</Math> about (8, -2) and its image is translated <Math>\begin{pmatrix} 3 \\ 0 \\ \end{pmatrix}</Math>. What single transformation is equivalent will have the same result?$

Shape I is enlarged by s.f. $\frac{1}{2}$ about (8, -2) and its image is translated $\begin{pmatrix} 3 \\ 0 \\ \end{pmatrix}$ . What single transformation is equivalent will have the same result?

An enlargement scale factor $-\frac{1}{2}$ about (5, 2)

An enlargement scale factor $\frac{1}{2}$ about (8, 4) ✓

An enlargement scale factor $-\frac{1}{2}$ about (8, 4)

A translation by $\begin{pmatrix} -1.5 \\ 0 \\ \end{pmatrix}$

A translation by $\begin{pmatrix} 3 \\ 0 \\ \end{pmatrix}$

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Lesson Details

Key learning points

By understanding what changes and what is invariant, you can determine whether a transformation has occurred.
Sometimes you might need to persevere in order to find the right transformation(s).
You may be able to check your deductions by carrying out the transformation.

Common misconception

There is only one way to describe what has happened to an object to create its image.

There may be multiple transformations or combinations of transformations that map the object to the image.

Keywords

Object - The object is the starting figure, before a transformation has been applied.
Image - The image is the resulting figure, after a transformation has been applied.
Invariant - A property of a shape is invariant if that property has not changed after the shape is transformed.
Invariant point - A point on a shape is invariant if that point has not changed location after the shape is transformed.