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Compose two plane transformations in order

Start a set of 8
Step 9 of 9

A point p is transformed by two successive steps. Step 1 acts first, followed by Step 2

Step 1: apply a dilation from the y-axis by factor 3, written as \(p' = \begin{pmatrix}3 & 0 \\ 0 & 1\end{pmatrix}p\). This map fixes the origin

Step 2: apply a translation by vector (1, 2), written as \(p' = p + \begin{pmatrix}1 \\ 2\end{pmatrix}\)

Use the witness point \(p=\left(-1,-2\right)\). Find the composite matrix and composite translation vector separately, then apply them to the witness point

(a)1 mark

Select the statement that matches the two-step composition and its reverse

(b)4 marks

Find the matrix part of the composite. A translation contributes the identity matrix to this calculation

\(M_2M_1\)

Rows, in order: \(1\), \(2\)

Columns, in order: \(1\), \(2\)

\[M_2M_1 = \begin{bmatrix} ? & ? \\ ? & ? \end{bmatrix}\]
Composite matrix
(c)2 marks

Find the translation vector of the composite. A matrix transformation contributes the zero translation vector

Transform the first step's translation vector by the second step's matrix, then include the second step's translation vector

(d)1 mark

Apply the composite to the witness point \(p=\left(-1,-2\right)\). Give its x-coordinate

(e)1 mark

Apply the composite to the witness point \(p=\left(-1,-2\right)\). Give its y-coordinate

Year 11, VCE Units 1 & 2, Chapter 7, 4 difficulty levels, No calculatorBuild a worksheet

Plane transformations

Compose operations in their stated order and reverse that order when undoing them

  1. Column convention The columns of A are the images of the standard basis columns
  2. Order and centre The last operation is the leftmost factor. Subtract a centre before rotating and add it afterwards
  3. Check Verify a point, inverse, fixed-point equation and nonnegative area factor as appropriate

Worked example

Reflect in the y-axis, then apply the shear (x,y)↦(x+2y,y)

\[\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}-1&0\\0&1\end{pmatrix}=\begin{pmatrix}-1&2\\0&1\end{pmatrix}\]
  1. \(\displaystyle (x,y)\mapsto(-x+2y,y)\)The reflection acts first; verify by substituting an arbitrary point

Common mistakes

Product order
✕First A, then B gives AB✓First A, then B gives BA
BAp means B(Ap)

Difficulty levels

introRead and apply a map
standardConstruct and calculate exact transformations
challengeRespect centres and composition order
examInterpret inverse, invariant and area conditions

Questions about this topic

Are fixed points the same as invariant sets?+

A fixed point maps to itself. An invariant set maps onto itself, but its points may move

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 7: Trigonometry and transformations

In this chapter

Convert angle measure and use the radian arc relationArea of a sectorUse the sine rule in a non-right-angled triangleUse the cosine rule in a non-right-angled triangleTrigonometry in three dimensionsExact circular-segment area and perimeterChoose perpendicular sections and calculate spatial anglesUse exact trigonometric identities and quadrant informationCompound, double-angle and reciprocal identitiesTransform and reconstruct sine and cosine functionsCombine waves and convert trigonometric sums and productsMap or recover points and finite coordinate setsCompose two plane transformations in orderRotate, reflect and interpret fixed points and area