1 of 8 0-day streak

Compose two plane transformations in order

Question 1 of 8

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

Step 1: apply a reflection in the x-axis, written as \(p' = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}p\). This map fixes the origin

Step 2: 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

Use the witness point \(p=\left(-1,-3\right)\). Find the composite matrix and composite translation vector separately, then apply them to the witness point

(a)1 mark

Which option gives the chronological order of the two steps?

Apply a dilation from the y-axis by factor 3, then a reflection in the x-axis

Apply a reflection in the x-axis, then a dilation from the y-axis by factor 3

(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

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

x-component y-component

(d)1 mark

Apply the composite to the witness point \(p=\left(-1,-3\right)\). Give its x-coordinate

x-coordinate

(e)1 mark

Apply the composite to the witness point \(p=\left(-1,-3\right)\). Give its y-coordinate

y-coordinate