Describing Transformations

Steps

1. 1

   Different size? Enlargement.

2. 2

   Same size, flipped over? Reflection.

3. 3

   Same size, turned? Rotation.

4. 4

   Same size, same way up, just moved? Translation.

Worked example

1. 1

   $\text{Rotation}$ — The shape is the same size and has only been turned, not flipped. A turn is a rotation.

2. 2

   $90^{\circ }\text{ clockwise}$ — The side that stood upright now lies flat. That is a quarter turn the way a clock’s hands go.

3. 3

   $\text{centre }(0,0)$ — The centre is the one point that stays still while the shape turns. Here it is the origin.

4. 4

   $(x,y)\to (y,-x):(1,4)\to (4,-1)$ — Check with one corner. A clockwise quarter turn about the origin swaps the numbers and changes the sign of the new second one, so (1, 4) lands on (4, −1).

Worked example

1. 1

   $A(3,3)\to A^{\prime }(5,3)$ — Take a corner and its image. They have the same y value, so the line joining them is flat.

2. 2

   $x=\frac{3+5}{2}=4$ — The line at right angles through the midpoint is upright, and its x value is halfway between 3 and 5.

3. 3

   $B(1,-1)\to B^{\prime }(1,5)$ — Take a second corner and its image. They have the same x value, so the line joining them is upright.

4. 4

   $y=\frac{-1+5}{2}=2$ — The line at right angles through the midpoint is flat, and its y value is halfway between −1 and 5: (−1) + 5 = 4 and 4 ÷ 2 = 2.

5. 5

   $\text{centre }(4,2)$ — The lines x = 4 and y = 2 cross at (4, 2). That is the centre of the rotation.

6. 6

   $A:(-1,1)\to A^{\prime }:(1,1)$ — Measure from the centre (4, 2). A is 1 left and 1 up. A′ is 1 right and 1 up. The two arms are at right angles, and the swing from up-left to up-right is clockwise.

7. 7

   $\text{Rotation }90^{\circ }\text{ clockwise, centre }(4,2)$ — Give the name, the angle with its direction, and the centre.

Worked example

1. 1

   $\text{Enlargement}$ — The image has the same shape but a different size, so it is an enlargement.

2. 2

   $\frac{2}{1}=2$ — Measure a matching pair of sides: 1 long on the object and 2 long on the image. Image ÷ object gives the scale factor.

3. 3

   $(2,3)\to (1,2)\to (0,1)$ — Draw a line through a corner and its image, and carry it back past the object. Each step back is 1 left and 1 down, and the line reaches (0, 1).

4. 4

   $(4,3)\to (2,2)\to (0,1)$ — Do the same with a second corner and its image. Each step back is 2 left and 1 down, and this line also reaches (0, 1).

5. 5

   $\text{centre }(0,1)$ — Both lines pass through (0, 1). Lines through matching corners always meet at the centre.

6. 6

   $(1,2)-(0,1)=(1,1),(2,3)-(0,1)=(2,2)$ — Check: the image corner is twice as far from the centre as the object corner, which agrees with scale factor 2.

7. 7

   $\text{Enlargement, scale factor }2,\text{centre }(0,1)$ — Give the name, the scale factor and the centre. Each one is a mark.

Worked example

1. 1

   $(2,1)\to (-4,1)$ — Take a corner and its image. The y value stays at 1, so the shape was flipped sideways and the mirror line is upright.

2. 2

   $\frac{2+(-4)}{2}=-1$ — The mirror line is exactly halfway between a corner and its image. Add the x values and halve: 2 + (−4) = −2, and −2 ÷ 2 = −1.

3. 3

   $\text{Reflection in }x=-1$ — Give the name and the equation of the mirror line. An upright line has an equation that starts x =.

4. 4

   $(1,2)\to (5,-1)$ — Now a different shape that has slid. Every corner moves the same way, so one corner shows the whole move.

5. 5

   $(5-1,-1-2)=(4,-3)$ — Take the object corner away from its image. Across: 5 − 1 = 4. Up: (−1) − 2 = −3.

6. 6

   $\text{Translation by }\left(\begin{array}{c}4\\ -3\end{array}\right)$ — Give the name and the column vector. The top number is across and the bottom is up, so −3 means 3 down.

Worked example

1. 1

   $(1,4)\to (4,1)$ — Take a corner and its image. Both numbers have changed, so the mirror line is not flat or upright.

2. 2

   $(1,4)\to (4,1)$ — The two numbers have swapped places. Swapping x and y is what a reflection in the line y = x does.

3. 3

   $\left(\frac{1+4}{2},\frac{4+1}{2}\right)=(2.5,2.5)$ — Check the midpoint. Its x and y values are equal, so it lies on y = x, as a mirror line must.

4. 4

   $\text{Reflection in }y=x$ — Give the name and the equation of the mirror line.

5. 5

   $(2,5)\to (-5,-2)$ — Now another corner and image. The numbers have swapped again, but both signs have changed too.

6. 6

   $\left(\frac{2+(-5)}{2},\frac{5+(-2)}{2}\right)=(-1.5,1.5)$ — Find the midpoint. Its y value is the opposite of its x value, so it lies on y = −x.

7. 7

   $\text{Reflection in }y=-x$ — Swap and change both signs: the mirror line is y = −x.