Worksheets · Precalculus

Transformations with matrices worksheet for Precalculus

Precalculus treats a transformation matrix as a function on the whole plane. A figure goes in as one vertex matrix and comes out with a single product, and the questions run both ways: build the matrix from a description, or name what a given matrix does, including dilations and translations done by adding.

Also taught in Algebra 2, at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Transformations with Matrices

Date Period

Answer each question.

  1. Transform a figure with matrices

    Rotate △ABC 90∘ counterclockwise about the origin:
    A(−3,1), B(2,−7), C(2,3)
  2. Rotate △ABC 90∘ clockwise about the origin:
    A(−8,4), B(−1,3), C(−7,8)
  3. Rotate quadrilateral ABCD 180∘ about the origin:
    A(1,4), B(−5,−5), C(3,−8), D(5,−2)
  4. Rotate △ABC 90∘ counterclockwise about the origin, then rotate it 180∘ about the origin:
    A(−7,−2), B(−5,−5), C(4,5)
  5. Reflect △ABC across the line y=−x, then reflect it across the line y=x: A(6,−5), B(2,4), C(−2,−7)

  1. Name the transformation

    The vertex matrix [ −18−14−41 ] becomes [ −18−1−44−1 ]. Name the transformation.
  2. Describe the transformation [ 0−110 ] performs.
  3. The vertex matrix [ 5−332−3−8 ] becomes [ 10−664−6−16 ]. Name the transformation.
  4. The vertex matrix [ 38−6−1−15 ] becomes [ 1540−30−5−525 ]. Name the transformation.
  5. Describe the transformation [ −100−1 ] performs.

  1. Write the transformation matrix

    Write the matrix for a reflection across the y-axis.
  2. Write the matrix for a reflection across the line y=x, followed by a reflection across the x-axis.
  3. Write the matrix for a dilation by a factor of 3.
  4. Write the matrix for a dilation by a factor of 12.
  5. Write the matrix for a dilation by a factor of 4.

  1. Translate with matrix addition

    Use matrix addition to translate triangle A(−6,4), B(5,4), C(−1,6) 4 left and 5 up.
  2. Use matrix addition to translate triangle A(−2,2), B(−5,−4), C(3,6) 1 right and 5 up.
  3. Use matrix addition to translate triangle A(6,−5), B(−6,−5), C(4,−3) 4 left and 3 down.
  4. Use matrix addition to translate triangle A(2,−4), B(4,5), C(1,2) 2 right and 4 down.
  5. Use matrix addition to translate triangle A(−3,−4), B(−5,2), C(3,6) 3 right and 4 up.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Transformations with MatricesAnswer keyVersion 2

Date Period

Answer each question.

  1. Transform a figure with matrices

    Rotate △ABC 90∘ counterclockwise about the origin:
    A(−3,1), B(2,−7), C(2,3)    A′(−1,−3), B′(7,2), C′(−3,2)
  2. Rotate △ABC 90∘ clockwise about the origin:
    A(−8,4), B(−1,3), C(−7,8)    A′(4,8), B′(3,1), C′(8,7)
  3. Rotate quadrilateral ABCD 180∘ about the origin:
    A(1,4), B(−5,−5), C(3,−8), D(5,−2)    A′(−1,−4), B′(5,5), C′(−3,8), D′(−5,2)
  4. Rotate △ABC 90∘ counterclockwise about the origin, then rotate it 180∘ about the origin:
    A(−7,−2), B(−5,−5), C(4,5)    A′(−2,7), B′(−5,5), C′(5,−4)
  5. Reflect △ABC across the line y=−x, then reflect it across the line y=x: A(6,−5), B(2,4), C(−2,−7)A′(−6,5), B′(−2,−4), C′(2,7)

  1. Name the transformation

    The vertex matrix [ −18−14−41 ] becomes [ −18−1−44−1 ]. Name the transformation.Reflection across the x-axis
  2. Describe the transformation [ 0−110 ] performs.Rotation of 90∘ counterclockwise
  3. The vertex matrix [ 5−332−3−8 ] becomes [ 10−664−6−16 ]. Name the transformation.Dilation by a factor of 2
  4. The vertex matrix [ 38−6−1−15 ] becomes [ 1540−30−5−525 ]. Name the transformation.Dilation by a factor of 5
  5. Describe the transformation [ −100−1 ] performs.Rotation of 180∘

  1. Write the transformation matrix

    Write the matrix for a reflection across the y-axis.[ −1001 ]
  2. Write the matrix for a reflection across the line y=x, followed by a reflection across the x-axis.[ 01−10 ]
  3. Write the matrix for a dilation by a factor of 3.[ 3003 ]
  4. Write the matrix for a dilation by a factor of 12.[ 120012 ]
  5. Write the matrix for a dilation by a factor of 4.[ 4004 ]

  1. Translate with matrix addition

    Use matrix addition to translate triangle A(−6,4), B(5,4), C(−1,6) 4 left and 5 up.A′(−10,9), B′(1,9), C′(−5,11)
  2. Use matrix addition to translate triangle A(−2,2), B(−5,−4), C(3,6) 1 right and 5 up.A′(−1,7), B′(−4,1), C′(4,11)
  3. Use matrix addition to translate triangle A(6,−5), B(−6,−5), C(4,−3) 4 left and 3 down.A′(2,−8), B′(−10,−8), C′(0,−6)
  4. Use matrix addition to translate triangle A(2,−4), B(4,5), C(1,2) 2 right and 4 down.A′(4,−8), B′(6,1), C′(3,−2)
  5. Use matrix addition to translate triangle A(−3,−4), B(−5,2), C(3,6) 3 right and 4 up.A′(0,0), B′(−2,6), C′(6,10)

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

How to do these

Write the matrix for a reflection across y=x, then use it on A(2,−1), B(4,3), C(−1,5)

  1. Reflecting across y = x sends (1, 0) to (0, 1) and (0, 1) to (1, 0), so the matrix has rows 0, 1 and 1, 0.
  2. Multiply it by the vertex matrix with columns (2, -1), (4, 3), (-1, 5): each column has its two entries swapped.

The answer is [ 0110 ]; A′(−1,2), B′(3,4), C′(5,−1).

Where students go wrong

Trying to translate with a 2x2 multiplication. No matrix product can move the origin, so a shift has to be done by adding a matrix with the shift repeated in every column.

Also called matrix transformations, reflection matrix, rotation matrix, dilation matrix or transformation matrices.

What you can put on this worksheet

Transform a figure with a matrix
Reflect △ABC across the x-axis:
A(−2,2), B(−2,−5), C(4,1) → A′(−2,−2), B′(−2,5), C′(4,−1)
Name the transformation a matrix performs
The vertex matrix [ 243315 ] becomes [ 243−3−1−5 ]. Name the transformation. → Reflection across the x-axis
Write the matrix for a transformation
Write the matrix for a reflection across the line y=x. → [ 0110 ]
Translate with matrix addition
Use matrix addition to translate triangle A(−3,6), B(−2,5), C(1,6) 4 left and 4 down. → A′(−7,2), B′(−6,1), C′(−3,2)
Dilation by a decimal factor
Use matrices to dilate the triangle with vertex matrix [ 4−701−48 ] by a factor of 1.5 → [ 6−10.501.5−612 ]

Questions about these worksheets

Yes. Every transformations with matrices 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.

Precalculus is usually taken around 11th or 12th 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 with matrices 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 5: transform a figure with a matrix, name the transformation a matrix performs, write the matrix for a transformation, translate with matrix addition and dilation by a decimal factor. Tick as many as you want and set how many of each, or let it spread them evenly.

Multiplying in the wrong order. The transformation matrix goes on the left and the vertices on the right; the other way round usually does not even multiply. When two transformations are done one after the other, the second one goes further left.