Combined Transformations

Steps

1. 1

   Do the first move. Label the new shape B.

2. 2

   Do the second move to B. Label the result C.

3. 3

   If asked, compare the first shape with C and describe one move that does the same.

Worked example

1. 1

   $(1,2)\to (1,-2)$ — Reflect A in the x-axis first. A corner flips to the other side of the axis, so the y value changes sign and the x value stays. Call the new shape B.

2. 2

   $(1,-2)\to (-1,-2)$ — Now reflect B in the y-axis. The x value changes sign and the y value stays. Call the result C.

3. 3

   $(1,2)\to (-1,-2)$ — Compare A with C only and ignore B. Both numbers have changed sign.

4. 4

   $\text{Rotation }180^{\circ },\text{centre }(0,0)$ — Both signs changing means a half turn about the origin. A half turn needs no direction.

Worked example

1. 1

   $(2,1)\to (1,2)$ — Order 1, move 1: reflect in y = x. This swaps the two numbers.

2. 2

   $(1,2)\to (2,-1)$ — Order 1, move 2: turn 90° clockwise about O. A clockwise quarter turn changes (x, y) to (y, −x), so (1, 2) goes to (2, −1).

3. 3

   $(2,1)\to (1,-2)$ — Order 2, move 1: turn 90° clockwise first. Using (x, y) → (y, −x), the point (2, 1) goes to (1, −2).

4. 4

   $(1,-2)\to (-2,1)$ — Order 2, move 2: reflect in y = x. This swaps the numbers, so (1, −2) goes to (−2, 1).

5. 5

   $(2,-1)\ne (-2,1)$ — The two orders end at different points, so the order matters.

Worked example

1. 1

   $(2,3)\to (0,3)$ — Reflect in x = 1 first. The point is 1 to the right of the line, so its image is 1 to the left, at x = 0.

2. 2

   $(0,3)\to (8,3)$ — Now reflect this image in x = 4. It is 4 to the left of that line, so the new image is 4 to the right, at x = 8.

3. 3

   $(8-2,3-3)=(6,0)$ — Compare the start with the end. The point has moved 6 right and 0 up.

4. 4

   $\text{Translation by }\left(\begin{array}{c}6\\ 0\end{array}\right)$ — Give the name and the column vector. The lines are 4 − 1 = 3 apart, and the slide is twice the gap: 2 × 3 = 6, from the first line towards the second.

Worked example

1. 1

   $\text{angle between the lines}=45^{\circ }$ — Reflect in the x-axis, then in y = x. The line y = x goes 1 across for every 1 up, so it makes 45° with the x-axis.

2. 2

   $2\times 45^{\circ }=90^{\circ }$ — The angle of rotation is twice the angle between the lines.

3. 3

   $\text{centre }(0,0)$ — The x-axis and y = x cross at the origin, so the origin is the centre.

4. 4

   $\text{from the }x\text{-axis to }y=x:\text{anticlockwise}$ — Going from the first line to the second means turning anticlockwise, so the rotation is anticlockwise.

5. 5

   $(2,1)\to (2,-1)\to (-1,2)$ — Check with a point. The x-axis gives (2, −1). Swapping the numbers for y = x gives (−1, 2).

6. 6

   $(x,y)\to (-y,x):(2,1)\to (-1,2)$ — A 90° anticlockwise turn about the origin does exactly this, so the check agrees.

7. 7

   $\text{Rotation }90^{\circ }\text{ anticlockwise, centre }(0,0)$ — Give the name, the angle with its direction, and the centre.