Geometric Transformations to Practice Basic Skills and Introduce Fundamental Concepts

IntGeoTrans_lead1This post shows how learning to apply geometric transformation rules to slide, reflect, rotate or resize a figure can benefit students. Geometric transformation graphing activities can help students learn important math concepts in the following ways:

  • Learn how to interpret a symbolic description of a geometric transformation rule to find the image point of a preimage point.
  • Learn what it means to reflect a figure over a line.
  • Learn what it means to slide or translate a figure.
  • Learn what it means to rotate a figure about a point.
  • Learn what it means to stretch or shrink a figure.
  • The idea that preimage and image points is equivalent to the idea of function inputs and outputs.
  • Worthwhile practice with the basic operations of signed numbers.
  • Worthwhile practice plotting points and drawing geometric transformation images.
  • A tacit introduction to the important mathematical concepts of function, inverse of a function and composition of functions.

This post does not discuss the general theory of affine transformations nor does it discuss the study of geometry from a geometric transformation point of view. For a general discussion of 2D matrix based geometric transformations, download my free handout Matrix Geometric Transformations or visit our free instructional content page.

The table below describes the geometric transformations considered in this post. Assume constants j and k are positive.

Transformation Operation
Reflect point (x,y) over the x-axis. (x,y) → (x,-y)
Reflect point (x,y) over the y-axis. (x,y) → (-x,y)
Reflect point (x,y) over the line y = x. (x,y) → (y,x)
Translate or slide point (x,y) right/left j units and up/down k units. (x,y) → (x ± j, y ± k)
Rotate point (x,y) 90° CW about (0,0). (x,y) → (y,-x)
Rotate point (x,y) 90° CCW about (0,0). (x,y) → (-y,x)
Rotate point (x,y) 180° about (0,0). (x,y) → (-x,-y)
Expand or contract point (x,y) by a factor of k from (0,0). (x,y) → (kx,ky)

The setup and parameters for a geometric transformation activity are shown below:

  1. Each student is provided a handout containing directions for the activity, an x-y coordinate axes with the graph of the preimage polygon drawn in black, a symbolic description of a geometric transformation, a table for the preimage points, and blank table for the image points to be filled in by the student. I prefer lattice point coordinate axes, but grid line coordinate axes are fine. The coordinate axes should be properly labeled and laid out so that it’s easy to plot points. It’s important that the coordinate axes be drawn with a 1:1 aspect ratio so that perpendicular lines appear to be perpendicular and graphs of circles appear to be circles, not ovals.
  2. Each student should have a ruler to aid in drawing graphs. Sloppy hand drawn graphs are not allowed. I have found that a 6 inch or 15 cm ruler works best.
  3. It is assumed that students can do the basic operations with signed numbers and plot points.
  4. No calculators allowed; strictly old school. Students may use scratch paper of course.
  5. The initial preimage point is arbitrary; just move from vertex to vertex around the polygon in either a clockwise or counterclockwise direction.
  6. To enhance the visual effect, allow colored ink pens or pencils to draw the image figures.

The tables and graphs below show the results of reflecting the same black preimage figure over the x-axis, the line x = -4 and the line y = x. Normally a geometric transformation graphing activity should have no more than two transformations to perform on a figure, but to conserve space three transformation images are graphed on the same x-y coordinate axes. The tables of x-y coordinates of the image points are color coded to match the color of the image polygon. In an actual transformation activity handout for students, the column of image point x-y coordinates is blank. The first two or three rows of the tables may show both preimage and image point coordinates to help students understand how the transformation rule works. The reflection over the line x = -4 is accomplished by chaining together transformations as follows:

  1. Slide the polygon right 4 units.
  2. Reflect the image over the y-axis.
  3. Slide the last image left 4 units.




The next demonstration involves a slide, rotation and size transformations of a preimage polygon drawn in black. The 90° counterclockwise rotation about (-2, 1) is accomplished by chaining together three transformations as follows:

  1. Slide the polygon right 2 units and down 1 unit.
  2. Rotate the image 90° counterclockwise about (0, 0).
  3. Slide the last image left 2 units and up 1 unit.

The image polygon drawn in green was obtained by chaining together a size transformation with expansion factor = 1.5 and a 180° rotation about (0, 0).




The graph below shows the decomposition of the geometric transformation (x, y) → (-y + 3 , x – 13) that rotates the black flag 90° counterclockwise about the point (8, – 5). Using the transformation mapping functions in the table above and the graphs of the 4 flags below, we can see that the geometric mapping function can be created as follows: (x, y) → (x – 8, y + 5) → (-(y + 5), x – 8) → (-(y + 5) + 8, x – 8 – 5) = (-y + 3, x – 13). Note that the transformations (x, y) → (x – 8, y +5) and (x, y) → (x + 8, y – 5) are inverse transformations.


Suggestions for fun follow up activities relating to geometric transformations:

  • After graphing the reflection image of a polygon over a line, have students fold the sheet of graph paper on the reflecting line, and then hold the sheet of graph paper up to the light to verify that preimage and image polygons are congruent.
  • After translating a polygon, use the theorem of Pythagoras to determine the magnitude of the slide and a protractor to determine the polar direction of the slide where the polar direction ranges from 0° to 360°.
  • Have students use a compass and protractor to verify that a polygon has been rotated a certain number of degrees about a point.
  • Let students be creative by having them make up their own transformation rule, and then use the rule to graph the image of a preimage polygon. Exceptional work can be posted in the classroom for all to enjoy.
  • Give students a graph similar to the graph above, and have them find a geometric transformation rule that maps a preimage to an image. You can let the preimage be any of the figures in the graph, and the image can be any of the other figures. You will be amazed to see that some students struggle to find a transformation rule that maps a given preimage figure onto itself, but this can be a great teaching opportunity!
  • Given the graphs of a preimage polygon and the image polygon under a size transformation, find the lengths of corresponding side pairs and verify that the ratios of the lengths of corresponding side pairs are equal.
  • Tell students that geometric transformations make it possible for game developers to create whose wonderful video games they love to play.

Here’s some exercises based on the examples in this post that you can give to your students:

Introduction to Geometric Transformations (student version)

Introduction to Geometric Transformations (teacher version)

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