A **transformation** is a process that manipulates a polygon or other two-dimensional object on a plane or coordinate system. Mathematical transformations describe how two-dimensional figures move around a plane or coordinate system.

A **preimage** or inverse image is the two-dimensional shape before any transformation. The **image** is the figure after transformation.

- Transformations Math Definition
- Types of Transformations
- Rigid and Non-Rigid Transformations
- Rigid and Non-Rigid Transformations
- Transformations Examples
- Transformations in the Coordinate Plane

There are five different transformations in math:

**Dilation**-- The image is a larger or smaller version of the preimage; "shrinking" or "enlarging."**Reflection**-- The image is a mirrored preimage; "a flip."**Rotation**-- The image is the preimage rotated around a fixed point; "a turn."**Shear**-- All the points along one side of a preimage remain fixed while all other points of the preimage move parallel to that side in proportion to the distance from the given side; "a skew.,"**Translation**-- The image is offset by a constant value from the preimage; "a slide."

Dilate a preimage of any polygon is done by duplicating its interior angles while increasing every side proportionally. You can think of dilating as resizing. Which triangle image, yellow or blue, is a dilation of the orange preimage?

The yellow triangle, a dilation, has been enlarged from the preimage by a factor of 3.

Imagine cutting out a preimage, lifting it, and putting it back face down. That is a reflection or a flip. A reflection image is a mirror image of the preimage. Which trapezoid image, red or purple, is a reflection of the green preimage?

The purple trapezoid image has been reflected along the x-axis, but you do not need to use a coordinate plane's axis for a reflection.

Using the origin, $(0,0)$, as the point around which a two-dimensional shape rotates, you can easily see rotation in all these figures:

A figure does not have to depend on the origin for rotation.

Here is a square preimage. To shear it, you "skew it," producing an image of a rhombus:

When a figure is sheared, its area is unchanged. A shear does not stretch dimensions; it does change interior angles.

A translation moves the figure from its original position on the coordinate plane without changing its orientation. Which octagon image below, pink or blue, is a translation of the yellow preimage?

The blue octagon is a translation, while the pink octagon has rotated.

A **rigid transformation** does not change the size or shape of the preimage when producing the image. Three transformations are rigid.

The rigid transformations are reflection, rotation, and translation. The image from these transformations will not change its size or shape.

A **non-rigid transformation** can change the size or shape, or both size and shape, of the preimage.

Two transformations, dilation and shear, are non-rigid. The image resulting from the transformation will change its size, its shape, or both.

There are five different types of transformations, and the transformation of shapes can be combined. A polygon can be reflected *and* translated, so the image appears apart and mirrored from its preimage. A rectangle can be enlarged and sheared, so it looks like a larger parallelogram.

Here are a preimage and an image. What *two* transformations were carried out on it?

The preimage has been rotated and dilated (shrunk) to make the image.

On a coordinate grid, you can use the x-axis and y-axis to measure every move. The lines also help with drawing the polygons and flat figures. Focus on the coordinates of the figure's vertices and then connect them to form the image.

Here is a tall, blue rectangle drawn in Quadrant $\mathrm{III}$.

We are asked to **translate** it to new coordinates. Mathematically, the graphing instructions look like this:

$(x,y)\to (x+9,y+5)$

This tells us to add $9$ to every $x$ value (moving it to the right) and add $9$ to every $Y$ value (moving it up):

$(-7,-1)\to (-7+9,-1+5)\to (2,4)$

Do the same mathematics for each vertex and then connect the new points in Quadrants $I$ and $\mathrm{IV}$.

Rotation using the coordinate grid is similarly easy using the x-axis and y-axis:

- To rotate $90\xb0$: $(x,y)\to (-y,x)$ (multiply the y-value times $-1$ and switch the x- and y-values)
- To rotate $180\xb0$: $(x,y)\to (-x,-y)$ (multiply both the y-value and x-value times $-1$)
- To rotate $270\xb0$: $(x,y)\to (y,-x)$ (multiply the x-value times $-1$ and switch the x- and y-values)

**Reflecting** a polygon across a line of reflection means counting the distance of each vertex to the line, then counting that same distance away from the line in the other direction. If the figure has a vertex at $(-5,4)$ and you are using the y-axis as the line of reflection, then the reflected vertex will be at $(5,4)$.

**Shearing** a figure means fixing one line of the polygon and moving all the other points and lines in a particular direction, in proportion to their distance from the given, fixed-line. Italic letters on a computer are examples of shear.

Mathematically, a shear looks like this, where *m* is the shear factor you wish to apply:

- $(x,y)\to (x+my,y)$ to shear $horizontally$
- $(x,y)\to (x,y+mx)$ to shear $vertically$

**Dilating** a polygon means repeating the original angles of a polygon and multiplying or dividing every side by a scale factor. If you have an isosceles triangle preimage with legs of $9feet$, and you apply a scale factor of $\frac{2}{3}$, the image will have legs of $6feet$.

In summary, a geometric transformation is how a shape moves on a plane or grid. Transformations, and there are rules that transformations follow in coordinate geometry.

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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