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

The image is the resulting figure, after a transformation has been applied.

A property of a shape is invariant if that property has not changed after the shape is transformed.

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

Advanced problem solving with further transformations

Maths

In this unit, pupils enhanced their understanding of enlargement. In this future unit, pupils investigate the effect of an enlargement on the perimeter and area of a 2-D shape and the volume of a 3-D shape. 

In the prior unit, pupils learnt about the different types of transformations. In this unit, pupils' knowledge is extended as negative scale factors are considered as are multiple transformations.

In this unit pupils apply their knowledge of transformations to explore negative enlargement factors and the application of multiple transformations.

Further transformations

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

Shape A is mapped onto shape B by a reflection. The equation of line that contains the invariant points is {{}}.

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 $$\begin{pmatrix} 3 \\ -4 \\ \end{pmatrix}$$

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?

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 $$(2, -3)$$

Jacob enlarges an object by a scale factor of $$-2$$ about $$(2, -3)$$. What transformation will map the image back onto the object?

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

Some Oak pupils are discussing invariant points after a transformation. Which pupils are correct?

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 $$\begin{pmatrix} 3 \\ 0 \\ \end{pmatrix}$$ 

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?

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

Select the statements which fully describe a transformation.

An enlargement scale factor -1 about (3, 0)

What type of transformation maps shape B onto shape A?

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

A translation by the vector $$\begin{pmatrix} 2 \\ 0 \\ \end{pmatrix}$$

Describe one possible transformation from A to B that has a collection of invariant points.

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 H onto shape E.

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)

Select the two transformations needed to map shape B onto shape G.

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