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Transformations, SMB-024

Points `P` and `Q`, shown on the Cartesian plane diagram, are rotated 180° about the origin and become points `P^(′)` and `Q^(′)`.

 

Plot the points `P^(′)` and `Q^(′)` on the diagram.  (3 marks)

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`P^(′)(2,1)`

`Q^(′)(-4,2)`

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`text{Rotating 180° = Rotating 90° twice (in either direction)}`

`text{1st 90° rotation (clockwise):}`

`P(-2,-1)\ ->\ (-1,2)`

`Q(4,-2)\ ->\ (-2,-4)`
 

`text{2nd 90° rotation (clockwise):}`

`(-1,2)\ ->\ P^(′)(2,1)`

`(-2,-4) \ ->\ Q^(′)(-4,2)`

 

Transformations, SMB-023

Point `Q(3,1)` on the Cartesian plane is rotated 180° about the origin in a clockwise direction to become point `Q^(′)`.

What are the coordinates of `Q^(′)`.  (2 marks)

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`Q^(′)(-3,-1)`

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`text{Rotating 180° = Rotating 90° twice (in either direction)}`

`text{1st 90° rotation (clockwise):}`

`Q(3,1)\ ->\ (1,-3)`

`text{2nd 90° rotation (clockwise):}`

`(1,-3)\ ->\ Q^(′)(-3,-1)`

Transformations, SMB-022

Point `A(4,-3)` on the Cartesian plane is rotated 90° about the origin in a clockwise direction to become point `A^(′)`.

What are the coordinates of `A^(′)`.  (1 mark)

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`A^(′)(-3,-4)`

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`A^(′)(-3,-4)`

Transformations, SMB-021

Point `P(-3,-7)` on the Cartesian plane is rotated 90° about the origin in an anticlockwise direction to become point `P^(′)`.

What are the coordinates of `P^(′)`.  (1 mark)

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`P^(′)(7,-3)`

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`P^(′)(7,-3)`

Transformations, SMB-020

Points `P` and `Q`, shown on the Cartesian plane diagram, are rotated 90° about the origin in a clockwise direction to become points `P^(′)` and `Q^(′)`.

Plot the points `P^(′)` and `Q^(′)` on the diagram.  (2 marks)

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Show Worked Solution

Transformations, SMB-019

Points `A` and `B`, shown on the Cartesian plane diagram, are rotated 90° about the origin in an anticlockwise direction to become points `A^(′)` and `B^(′)`.

Plot the points `A^(′)` and `B^(′)` on the diagram.  (2 marks)

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Show Worked Solution

Transformations, SMB-015

Gabby put 5 points on a grid and labelled them `A` to `E`, as shown on the diagram below.

Point `A` is 35 millimetres from point `D.`

Gabby adds a sixth point, `F` so that the arrangement of points has one line of symmetry.

How far is point `F` from point `B?`  (3 marks)

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`42\ text{mm}`

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`A\ text(to)\ D\ text(is 35 mm)`

`=> 1\ text(grid is 7 mm × 7 mm)`
 

`text(Point)\ F\ text(is added below:)`

`:.\ text(Distance)\ B\ text(to)\ F`

`= 6 xx 7`

`= 42\ text(mm)`

Transformations, SMB-013

The trapezium `ABCD` is moved to the new position shown by trapezium `SRQP.`

Which of these transformations resulted in the new position?

  1. Rotate `ABCD` 180° clockwise about the origin.
  2. Rotate `ABCD` 270° clockwise about the origin.
  3. Reflect `ABCD` across the `x`-axis, then translate 8 units left.
  4. Reflect `ABCD` across the `y`-axis, then translate 7 units down.
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`C`

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`text(Reflection in the)\ xtext(-axis:)`

`ABCD -> A^{prime}B^{prime}C^{prime}D^{prime}`

`text(Translate 8 units left:)`

`A^{prime}B^{prime}C^{prime}D^{prime} -> SRQP`

`=>C`

Transformations, SMB-012 MC

Rochelle drew a pattern which is pictured below.

Rochelle rotates the pattern.

How much does Rochelle to turn the pattern until it looks exactly the same?

  1. `1/8\ text(turn)`
  2. `1/4\ text(turn)`
  3. `1/2\ text(turn)`
  4. `text(1 turn)`
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`B`

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`text(Outer pattern looks the same every)\ \ 1/8\ \ text(turn).`

`text(Inner cross pattern looks the same every)\ \ 1/4\ \ text(turn).`

`:.\ text(Whole pattern looks the same every)\ \ 1/4\ \ text(turn).`

`=>B`

Transformations, SMB-010

This shape will be translated 4 units to the right and 2 units up.

Where will the image of the point `A` be located after the shape is translated?  (2 marks)

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`(1,−3)`

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`text(Current position)\ A(−3,−5)`

`text(Translate 4 units to right:)`

`A(−3 + 4,−5) \ -> \ (1,−5)`
 

`text(Translate 2 units up:)`

`(1,−5+2 ) \ -> \ (1,−3)`
 

`:. text(Image of)\ A\ text(is)\ (1,−3)`

Transformations, SMB-009

The point  `P(-1, -4)`  lies on the Cartesian plane. It is reflected in the `x`-axis to form the point `P^(′)`.

Find the coordinates of `P^(′)`.  (1 mark)

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`(-1,4)`

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`text(Reflections in the)\ xtext{-axis:}`

`ytext{-coordinate has opposite sign and}\ xtext{-coordinate is the same.}`

`P(-1,-4)\ ->\ P^(′)(-1,4)`

Transformations, SMB-008

The point  `P(-3, 7)`  lies on the Cartesian plane. It is reflected in the `y`-axis to form the point `P^(′)`.

Find the coordinates of `P^(′)`.  (1 mark)

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`(7,5)`

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`text(Reflections in the)\ ytext{-axis:}`

`xtext{-coordinate has opposite sign and}\ ytext{-coordinate is the same.}`

`P(-3,7)\ ->\ P^(′)(3,7)`

Transformations, SMB-007

`P(2,3)` is translated 3 units up and 4 units left.

The new point is then reflected in `x`-axis to form point `P^(′)`.

Find the coordinates of `P^(′)`.  (2 marks)

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`(-2,-6)`

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`text{1st transformation:}`

`(2,3)\ ->\ (2-4, 3+3)\ ->\ (-2,6)`
 

`text{2nd transformation (reflection):}`

`(-2,6)\ ->\ P^(′)(-2,-6)`

Transformations, SMB-006

`P(-3,-5)` is reflected in the `x`-axis and then translated 3 units to the right to form point `P^(′)`.

Find the coordinates of `P^(′)`.  (2 marks)

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`(0,5)`

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`text{1st transformation (reflection):}`

`P(-3,-5)\ ->\ (-3,5)`
 

`text{2nd transformation:}`

`(-3,5)\ ->\ (0,5)`

Transformations, SMB-005

The point  `A(-2, 5)`  lies on the Cartesian plane. It is translated five units left and then reflected in the `y`-axis.

Find the coordinates of the final image of `A`.  (2 marks)

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`(7,5)`

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`text(1st transformation:)`

`A(-2, 5)\ ->\ (-7,5)`
 

`text{2nd transformation (reflection):}`

`(-7,5)\ ->\ (7,5)`

Transformations, SMB-004

The point  `P(4, -3)`  lies on the Cartesian plane. It is translated four units vertically up and then reflected in the `y`-axis.

Find the coordinates of the final image of `P`.  (2 marks)

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`(-4,1)`

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`text(1st transformation:)`

`P(4,-3)\ ->\ (4,1)`
 

`text{2nd transformation (reflection):}`

`(4,1)\ ->\ (-4,1)`

Transformations, SMB-001 MC

Point `Q` is translated down 3 units.

What are the new coordinates of  `Q`?

  1. `(4,5)`
  2. `(1,2)`
  3. `(7,2)`
  4. `(4,-1)`
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`D`

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`Q\ text(has coordinates)\ (4,2)`

`text(When translated 3 down, the)\ ytext(-value decreases by 3.)`

`:.\ text{New coordinates are  (4, −1)}.`

`=>D`