Translations, reflections and rotations with coordinates

P is three units left of the origin and two units above it. Which ordered pair locates P?

An image point keeps its correspondence

A transformation moves points according to a stated rule. The starting point is the original, also called the pre-image

P prime, written P′, names the image of P. A′ must correspond to A, not another nearby vertex

The first coordinate is horizontal and the second is vertical in both pairs

For a triangle, transform every vertex and keep its labels in order. Join corresponding image vertices to show the image shape

A translation slides every point by the same amount

A horizontal change affects x and a vertical change affects y

Right and up are positive changes; left and down are negative changes

For a movement four units right and two units down, add four to x and subtract two from y

No point is turned or reflected. Every side keeps its length and direction

If one change is zero, that coordinate stays the same

Translate across the vertical axis

Translate every point 4 units right and 2 units down

P prime names the image of P. All coordinates use the same one-unit scale

Complete every image coordinate pair. Enter whole numbers, including negative values or zero where needed

Translate every point 4 units right and 2 units down

For P, start this stage at (-2,3)

Keep each x-coordinate first and its paired y-coordinate second

Every point moves by the same signed amounts, so lengths and angles are unchanged

A translation preserves shape, size and orientation

Combine a leftward and upward movement

A horizontal slide leaves y unchanged

Translate every point 2 units right, keeping its vertical coordinate unchanged

P prime names the image of P. All coordinates use the same one-unit scale

Complete every image coordinate pair. Enter whole numbers, including negative values or zero where needed

Translate every point 2 units right, keeping its vertical coordinate unchanged

For P, start this stage at (-3,-1)

Keep each x-coordinate first and its paired y-coordinate second

Every point moves by the same signed amounts, so lengths and angles are unchanged

A translation preserves shape, size and orientation

A vertical slide leaves x unchanged

Apply one movement to all three vertices

Translate every point 2 units right and 3 units up

Keep the correspondence A to A prime, B to B prime and C to C prime. All coordinates use the same one-unit scale

Complete every image coordinate pair. Enter whole numbers, including negative values or zero where needed

Translate every point 2 units right and 3 units up

For A, start this stage at (-3,-2)

For B, start this stage at (-1,-2)

For C, start this stage at (-3,-1)

Keep each x-coordinate first and its paired y-coordinate second

Every point moves by the same signed amounts, so lengths and angles are unchanged

A translation preserves shape, size and orientation

Preserve the labels while shifting the triangle

The original triangle has A(−3, −2), B(−1, −2) and C(−3, −1). Change dx and dy and compare labelled A with A′ while every vertex follows the same movement. Zero changes return the image to the original position

At dx = 2 and dy = 3, which statement is correct?

Recover the movement from matched points

Subtract the original coordinate from the image coordinate in each direction

If P is at negative two, three and P′ is at two, one

The movement is four units right and two units down

For a set of points, check every matched pair. Different movements for different vertices do not form one translation

Subtract original from image, not the reverse

One translation takes every original point to its matched image

P prime names the image of P. All coordinates use the same one-unit scale

Find the signed horizontal change dx and vertical change dy from original to image. Enter whole numbers, including negative values or zero where needed

For dx and dy, positive means right or up and negative means left or down

A translation adds the same horizontal and vertical change to every point

Compare P with its matched image, not another vertex

All matched pairs give the same two signed changes

Every point moves by the same signed amounts, so lengths and angles are unchanged

A translation preserves shape, size and orientation

Confirm one translation using all three correspondences

Reflecting in the x-axis reverses the vertical position

The x-axis is a horizontal mirror line. Each image is the same perpendicular distance on its opposite side

The horizontal position stays fixed while the vertical coordinate changes sign

A point already on the x-axis has y equal to zero and stays fixed

A mirror reflection reverses the orientation of the labelled triangle without changing its side lengths or angles

Keep x and change the sign of y

Reflect every point in the x-axis

P prime names the image of P. All coordinates use the same one-unit scale

Complete every image coordinate pair. Enter whole numbers, including negative values or zero where needed

Reflect every point in the x-axis

For P, start this stage at (-2,3)

Keep each x-coordinate first and its paired y-coordinate second

Points on the mirror axis stay fixed; all other points change the sign of the coordinate perpendicular to that axis

Reflect a point below the horizontal mirror

A point on the mirror line stays where it is

Reflect every vertex in the same horizontal axis

Reflect every point in the x-axis

Keep the correspondence A to A prime, B to B prime and C to C prime. All coordinates use the same one-unit scale

Complete every image coordinate pair. Enter whole numbers, including negative values or zero where needed

Reflect every point in the x-axis

For A, start this stage at (1,1)

For B, start this stage at (3,1)

For C, start this stage at (1,2)

Keep each x-coordinate first and its paired y-coordinate second

Points on the mirror axis stay fixed; all other points change the sign of the coordinate perpendicular to that axis

This single reflection preserves shape and size but reverses triangle orientation

Keep an axis vertex fixed while reflecting the others

Reflecting in the y-axis reverses the horizontal position

The y-axis is a vertical mirror line. Keep the vertical coordinate and change the sign of the horizontal coordinate

A point on the y-axis has x equal to zero and stays fixed

The axis name does not name the coordinate to change. Reflection changes the coordinate perpendicular to the mirror axis

Change x and keep y

Reflect every point in the y-axis

P prime names the image of P. All coordinates use the same one-unit scale

Complete every image coordinate pair. Enter whole numbers, including negative values or zero where needed

Reflect every point in the y-axis

For P, start this stage at (-4,-2)

Keep each x-coordinate first and its paired y-coordinate second

Points on the mirror axis stay fixed; all other points change the sign of the coordinate perpendicular to that axis

Move to the opposite side of the vertical mirror

Preserve vertex correspondence through a vertical reflection

Reflect every point in the y-axis

Keep the correspondence A to A prime, B to B prime and C to C prime. All coordinates use the same one-unit scale

Complete every image coordinate pair. Enter whole numbers, including negative values or zero where needed

Reflect every point in the y-axis

For A, start this stage at (1,1)

For B, start this stage at (3,1)

For C, start this stage at (1,2)

Keep each x-coordinate first and its paired y-coordinate second

Points on the mirror axis stay fixed; all other points change the sign of the coordinate perpendicular to that axis

This single reflection preserves shape and size but reverses triangle orientation

A vertex on the y-axis stays fixed

A(2, 3) maps to A′(−2, 3), and B(4, 1) maps to B′(−4, 1). Which coordinate-axis reflection gives both correspondences?

A rotation needs a centre, an angle and a direction

Here the centre is always the origin. Imagine turning the plane around that fixed point

A quarter turn is ninety degrees and a half turn is one hundred and eighty degrees

Clockwise follows a clock hand. Anticlockwise turns the opposite way

A ninety-degree clockwise turn has the same result as a two-hundred-and-seventy-degree anticlockwise turn

At a half turn, clockwise and anticlockwise reach the same image

A rotation keeps each point the same distance from the origin and preserves the size, shape and orientation of a triangle

For a clockwise quarter turn, swap then negate the original x

The original vertical coordinate becomes the new horizontal coordinate. The negative of the original horizontal coordinate becomes the new vertical coordinate

Use the original x and y in the rule together. Do not overwrite one coordinate before using it to find the other

Rotate a point through a clockwise quarter turn

Rotate every point 90 degrees clockwise about the origin

P prime names the image of P. All coordinates use the same one-unit scale

Complete every image coordinate pair. Enter whole numbers, including negative values or zero where needed

Rotate every point 90 degrees clockwise about the origin

For P, start this stage at (-2,3)

Keep each x-coordinate first and its paired y-coordinate second

The origin stays fixed. The rule preserves distance from the origin as well as distances between points

A rotation preserves shape, size and orientation

Apply the quarter-turn rule to every triangle vertex

A point on an axis can move to another axis

For an anticlockwise quarter turn, swap then negate the original y

The negative of the original vertical coordinate becomes the new horizontal coordinate. The original horizontal coordinate becomes the new vertical coordinate

Clockwise and anticlockwise quarter turns generally give different images. State the direction, not only the angle

Keep the quarter-turn direction consistent

Rotate every point 90 degrees anticlockwise about the origin

P prime names the image of P. All coordinates use the same one-unit scale

Complete every image coordinate pair. Enter whole numbers, including negative values or zero where needed

Rotate every point 90 degrees anticlockwise about the origin

For P, start this stage at (3,-2)

Keep each x-coordinate first and its paired y-coordinate second

The origin stays fixed. The rule preserves distance from the origin as well as distances between points

A rotation preserves shape, size and orientation

Rotate a triangle with a vertex at the centre

A half turn changes both signs without swapping coordinates

Two quarter turns in the same direction make a half turn

The origin stays fixed because changing the sign of zero still gives zero

Do not confuse changing both signs with reflecting in one coordinate axis

Reverse both positions through the origin

Rotate every point 180 degrees about the origin

P prime names the image of P. All coordinates use the same one-unit scale

Complete every image coordinate pair. Enter whole numbers, including negative values or zero where needed

Rotate every point 180 degrees about the origin

For P, start this stage at (-4,2)

Keep each x-coordinate first and its paired y-coordinate second

The origin stays fixed. The rule preserves distance from the origin as well as distances between points

A rotation preserves shape, size and orientation

Keep the vertex order in a half-turned triangle

The rotation centre is a fixed point

Record the intermediate image before the second operation

For a sequence of transformations, the second operation acts on the result of the first

Reflecting in the x-axis and then in the y-axis gives

The final rule is the same as a half turn about the origin

For the point two, three

In the two-stage tables, record both the intermediate pair and the final pair. A correct final image alone does not complete both requested stages

Reflect successively in the two coordinate axes

First reflect in the x-axis

Then reflect the intermediate image in the y-axis

Keep the correspondence A to A prime, B to B prime and C to C prime. All coordinates use the same one-unit scale

Both stages for A are supplied as worked data. Complete the missing stages for B and C

Complete the missing intermediate and final coordinate pairs in the stated order. Enter whole numbers, including negative values or zero where needed

Stage 1: First reflect in the x-axis

For A, start this stage at (1,1)

For B, start this stage at (3,1)

For C, start this stage at (1,2)

Stage 2: Then reflect the intermediate image in the y-axis

For A, start this stage at (1,-1)

For B, start this stage at (3,-1)

For C, start this stage at (1,-2)

Keep each x-coordinate first and its paired y-coordinate second

The origin stays fixed. The rule preserves distance from the origin as well as distances between points

Two reflections in these perpendicular axes preserve shape and size and restore the original orientation

Keep the origin fixed through both reflections

Changing the order can change the final image

Start at P(1, 2). Compare a translation one unit right and two units up with reflection in the x-axis

Translation first, then reflection gives

Reflection first, then the same translation gives

The final images differ. Follow chronological order rather than applying every rule separately to the starting coordinates

Some sequences do agree, as the two axis reflections give a half turn. Test the stated operations rather than assuming all orders agree or all orders differ

Translate before reflecting the intermediate image

Translate every point 1 unit right and 2 units up

Then reflect the translated image in the x-axis

Keep the correspondence A to A prime, B to B prime and C to C prime. All coordinates use the same one-unit scale

Both stages for A are supplied as worked data. Complete the missing stages for B and C

Complete the missing intermediate and final coordinate pairs in the stated order. Enter whole numbers, including negative values or zero where needed

Stage 1: Translate every point 1 unit right and 2 units up

For A, start this stage at (1,1)

For B, start this stage at (3,1)

For C, start this stage at (1,2)

Stage 2: Then reflect the translated image in the x-axis

For A, start this stage at (2,3)

For B, start this stage at (4,3)

For C, start this stage at (2,4)

Keep each x-coordinate first and its paired y-coordinate second

Apply the second operation to the intermediate image, not to the original points

This single reflection preserves shape and size but reverses triangle orientation

Apply the mirror after a leftward and upward translation

Reflect before applying the same translation

First reflect in the x-axis

Then translate every point 1 unit right and 2 units up

Keep the correspondence A to A prime, B to B prime and C to C prime. All coordinates use the same one-unit scale

Both stages for A are supplied as worked data. Complete the missing stages for B and C

Complete the missing intermediate and final coordinate pairs in the stated order. Enter whole numbers, including negative values or zero where needed

Stage 1: First reflect in the x-axis

For A, start this stage at (1,1)

For B, start this stage at (3,1)

For C, start this stage at (1,2)

Stage 2: Then translate every point 1 unit right and 2 units up

For A, start this stage at (1,-1)

For B, start this stage at (3,-1)

For C, start this stage at (1,-2)

Keep each x-coordinate first and its paired y-coordinate second

Apply the second operation to the intermediate image, not to the original points

This single reflection preserves shape and size but reverses triangle orientation

Translate the reflected coordinates, not the originals

A triangle is reflected in the y-axis. Which statement must be true?

Repeated transformations create predictable patterns

Repeatedly translating by two units right keeps adding two to the horizontal coordinate

Repeated ninety-degree anticlockwise rotations around the origin return a point after four turns

Reflecting twice in the same axis returns every point to its starting position

Record each stage to reproduce a pattern reliably and to explain when a sequence returns to the original

Keep the movement, coordinates and correspondence consistent

Translate by adding the same signed horizontal and vertical changes to every point

Reflecting in the x-axis changes y; reflecting in the y-axis changes x

Quarter turns swap the coordinates and change one appropriate sign; a half turn changes both signs

For combinations, transform the intermediate image next. Order can matter

Keep vertex labels matched. These movements preserve shape and size, while a single reflection reverses orientation