Worksheets · Geometry

Transformations worksheet

Each transformation is one rule about what happens to the coordinates. Learn the six rules and the whole topic is arithmetic.

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Sigma Prep · sigmaprep.io/worksheets

Name

Transformations

Date Period

Find the image of each point or figure.

  1. Transform a figure on a grid

    Reflect the triangle across the y-axis.−10−10−5−5551010ABC
  2. Rotate 180∘ about the origin.−10−10−5−551010ABC5
  3. Draw the image of the triangle after a reflection in the x-axis.−10−10−5−5551010ABC
  4. Reflect the triangle across the x-axis.−10−10−5−5551010ABC
  5. Draw the image of the triangle after a reflection in the y-axis.−10−10−5−5551010ABC
  6. Rotate the triangle 180∘ around the origin.−10−10−5−5551010ABC
  7. Reflect over the y-axis.−10−10−5−5551010ABC

  1. Translate a point

    Translate (8,2) by (x,y)→(x+4,  y−5).
  2. Translate (−10,12) by (x,y)→(x+14,  y−13).
  3. Translate (−7,−5) by (x,y)→(x+14,  y+4).
  4. Translate (−2,−4) by (x,y)→(x+4,  y−6).
  5. Translate (9,−8) by (x,y)→(x−13,  y−3).
  6. Translate (−12,6) by (x,y)→(x−10,  y+12).
  7. Translate (8,3) by (x,y)→(x−8,  y−8).

  1. Find the image after a sequence of transformations

    Rotate 180∘ about the origin, then reflect over the x-axis, then translate (x,y)→(x+3,y−1). Where does (−1,−5) go?
  2. Rotate 180∘ about the origin, then translate (x,y)→(x+3,y+2), then translate (x,y)→(x−4,y−1). Where does (4,6) go?
  3. Reflect over the y-axis, then rotate 90∘ counterclockwise about the origin, then reflect over y=x. Where does (−3,−3) go?
  4. Reflect over the x-axis, then rotate 180∘ about the origin, then translate (x,y)→(x+3,y−1). Where does (5,3) go?
  5. Reflect over the y-axis, then reflect over y=x, then rotate 180∘ about the origin. Where does (1,−1) go?
  6. Reflect over the y-axis, then rotate 180∘ about the origin, then rotate 90∘ counterclockwise about the origin. Where does (−1,−3) go?

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

TransformationsAnswer keyVersion 1

Date Period

Find the image of each point or figure.

  1. Transform a figure on a grid

    Reflect the triangle across the y-axis.−10−10−5−5551010ABCA′B′C′(6,4),  (2,4),  (6,1)
  2. Rotate 180∘ about the origin.−10−10−5−551010ABCA′B′C′5(−4,1), (−4,5), (−7,1)
  3. Draw the image of the triangle after a reflection in the x-axis.−10−10−5−5551010ABCA′B′C′(3,−2), (3,−5), (7,−3)
  4. Reflect the triangle across the x-axis.−10−10−5−5551010ABCA′B′C′(−6,3), (−2,7), (−6,7)
  5. Draw the image of the triangle after a reflection in the y-axis.−10−10−5−5551010ABCA′B′C′(−3,6), (−6,5), (−3,3)
  6. Rotate the triangle 180∘ around the origin.−10−10−5−5551010ABCA′B′C′(−2,4), (−5,4), (−2,7)
  7. Reflect over the y-axis.−10−10−5−5551010ABCA′B′C′(7,3),  (5,7),  (4,4)

  1. Translate a point

    Translate (8,2) by (x,y)→(x+4,  y−5).(12,−3)
  2. Translate (−10,12) by (x,y)→(x+14,  y−13).(4,−1)
  3. Translate (−7,−5) by (x,y)→(x+14,  y+4).(7,−1)
  4. Translate (−2,−4) by (x,y)→(x+4,  y−6).(2,−10)
  5. Translate (9,−8) by (x,y)→(x−13,  y−3).(−4,−11)
  6. Translate (−12,6) by (x,y)→(x−10,  y+12).(−22,18)
  7. Translate (8,3) by (x,y)→(x−8,  y−8).(0,−5)

  1. Find the image after a sequence of transformations

    Rotate 180∘ about the origin, then reflect over the x-axis, then translate (x,y)→(x+3,y−1). Where does (−1,−5) go?(4,−6)
  2. Rotate 180∘ about the origin, then translate (x,y)→(x+3,y+2), then translate (x,y)→(x−4,y−1). Where does (4,6) go?(−5,−5)
  3. Reflect over the y-axis, then rotate 90∘ counterclockwise about the origin, then reflect over y=x. Where does (−3,−3) go?(3,3)
  4. Reflect over the x-axis, then rotate 180∘ about the origin, then translate (x,y)→(x+3,y−1). Where does (5,3) go?(−2,2)
  5. Reflect over the y-axis, then reflect over y=x, then rotate 180∘ about the origin. Where does (1,−1) go?(1,1)
  6. Reflect over the y-axis, then rotate 180∘ about the origin, then rotate 90∘ counterclockwise about the origin. Where does (−1,−3) go?(−3,−1)

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Reflect over the x-axis, then rotate 90∘ counterclockwise about the origin. Where does (3,−2) go?

  1. Over the x-axis the y changes sign: (3, 2).
  2. A quarter turn counterclockwise sends (x, y) to (-y, x): (-2, 3).

The answer is (−2,3).

Where students go wrong

Turning the wrong way, or swapping without changing a sign. A quick check: sketch the point roughly and see which quadrant it should land in. If the answer is in the wrong quarter of the plane, the rule was misremembered.

Also called coordinate transformations.

What you can put on this worksheet

On a grid, with the image drawn
Draw the image of the triangle after a reflection in the y-axis. → (5,3),  (9,3),  (7,6)
Translations
Translate (3,1) by (x,y)→(x+2,  y+2). → (5,3)
Reflections
Reflect (3,1) over the y-axis. → (−3,1)
Rotations
Rotate (3,1) 180∘ about the origin. → (−3,−1)
Name the transformation and write its rule
Write the rule (x,y)→(?,?) for the transformation shown, and name the transformation. → Reflection across the x-axis: (x,y)→(x,−y)
Over x = a, y = b and y = -x
The point (−6,0) is reflected across y=4. Give the coordinates of the image. → (−6,8)
Compositions
Translate (x,y)→(x−3,y−5), then reflect over the y-axis. Where does (4,1) go? → (−1,−4)
A segment or a quadrilateral, not a triangle
Rotate 180∘ about the origin. → (2,−3), (6,−3), (7,−6), (4,−7)

Questions about these worksheets

Yes. Every transformations sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Geometry is usually taken around 9th or 10th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the transformations questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 8: on a grid, with the image drawn, translations, reflections, rotations, name the transformation and write its rule, over x = a, y = b and y = -x, compositions and a segment or a quadrilateral, not a triangle. Tick as many as you want and set how many of each, or let it spread them evenly.

Turning the wrong way, or swapping without changing a sign. A quick check: sketch the point roughly and see which quadrant it should land in. If the answer is in the wrong quarter of the plane, the rule was misremembered.