Recovering a point from its image

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Before you start

Matrix transformations. Read the relevant method, then decide which parts you can solve without hints.

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Read an original example question

A transformation has matrix $\begin{pmatrix}0& −3\\1& 0\end{pmatrix}$ . It maps P to P′ = (6, 4).

Part a

Find the coordinates of P.

Part b

Describe this transformation as a rotation followed by a stretch.

Part c

Find the factor by which areas change.

Model solution and review criteria

Part a

For P = (x, y), the image is (−3y, x). Thus x = 4 and y = -2.

Write the two image equations.
Solve for the original coordinates.

Part b

A 90° anticlockwise rotation about the origin gives (−y, x), then a horizontal stretch of factor 3 gives (−3y, x).

State the rotation angle, centre and direction.
State the stretch factor and horizontal direction.

Part c

The rotation preserves area and the horizontal stretch multiplies it by 3. Equivalently, the determinant is 3 and area scales by its absolute value.

Calculate the determinant.
Use its absolute value for area.

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