Rotations, reflections and dilations | Year 11

01 / LEARN

The idea, step by step

Test a standard transformation

Dilation of factor k about the origin has matrix kI. Reflection in the x-axis keeps x and reverses y; reflection in $y = x$ swaps the two components. Test a point on the mirror line: it must stay fixed.

A rotation here is about the origin. To rotate about another centre, first translate that centre to the origin, rotate, then translate back.

A counterclockwise rotation

The image of the first unit vector is ${(cos θ, sin θ)}^{T}$ and that of the second is ${(- sin θ, cos θ)}^{T}$ . These become the columns of the rotation matrix. Negative theta describes clockwise rotation.

A coordinate calculation should agree with the direction on a sketch. In particular, a counterclockwise quarter-turn sends the positive x-axis to the positive y-axis.

R_{θ} = (\begin{array}{cccccccccc} cos θ & - sin θ \\ sin θ & cos θ \end{array})

A common trap

Changing the wrong coordinate sign for a reflection.

Instead: Check a point on the mirror line; it must stay fixed.

02 / WORKED EXAMPLES

See the thinking

Example 1

A quarter-turn

Rotate $(2, - 3)$ through $90^{\circ}$ counterclockwise.

The rule is (x,y) to (-y,x).
Substitute x=2 and y=-3.

(3, 2)

Example 2

A reflection and a dilation

Reflect $(1, 4)$ in $y = x$ , then dilate by factor 2.

Swap coordinates, obtaining (4,1).
Double both components.

(8, 2)

; combined matrix

(\begin{array}{cccccccccc} 0 & 2 \\ 2 & 0 \end{array})

Check your understanding

Give the matrix for a $180^{\circ}$ rotation.

Show answer

$(\begin{array}{cccccccccc} - 1 & 0 \\ 0 & - 1 \end{array})$ .

Both coordinate unit vectors reverse.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Give the matrix for reflection in the y-axis.
2 marks
Rotate $(1, 2)$ through $90^{\circ}$ clockwise.
2 marks
Give the matrix for dilation of factor -2.
2 marks
Reflect $(3, - 1)$ in $y = x$ .
2 marks
Rotate $(2, 0)$ through $60^{\circ}$ counterclockwise using exact values.
2 marks
Describe the transformation $(\begin{array}{cccccccccc} 0 & - 1 \\ - 1 & 0 \end{array})$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Give the matrix for reflection in the y-axis.
2 marks
Change the x sign but keep y.
$(\begin{array}{cccccccccc} - 1 & 0 \\ 0 & 1 \end{array})$ .
Rotate $(1, 2)$ through $90^{\circ}$ clockwise.
2 marks
Use (x,y) to (y,-x).
$(2, - 1)$ .
Give the matrix for dilation of factor -2.
2 marks
Multiply both coordinates by -2.
$(\begin{array}{cccccccccc} - 2 & 0 \\ 0 & - 2 \end{array})$ .
Reflect $(3, - 1)$ in $y = x$ .
2 marks
Swap the coordinates.
$(- 1, 3)$ .
Rotate $(2, 0)$ through $60^{\circ}$ counterclockwise using exact values.
2 marks
Use $2 cos 60^{\circ} = 1$ and $2 sin 60^{\circ} = \sqrt{3}$ .
$(1, \sqrt{3})$ .
Describe the transformation $(\begin{array}{cccccccccc} 0 & - 1 \\ - 1 & 0 \end{array})$ .
2 marks
It sends (x,y) to (-y,-x).
Every point (t,-t) on that mirror line is fixed.
Reflection in $y = - x$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: matrices and transformations; original lessons follow the approved school-sized chapter structure.

Official curriculum source ↗

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