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Finding the Image of a Composition & Comparing Orders of Compositions

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will give a tutorial on how to find the image of compositions of transformations. It will also demonstrate what can happen when the order of the transformations is changed.

Compositions of Transformations

Imagine that you ask your friend for directions to their house from your house. Your friend gives you turn-by-turn directions just as a GPS would. If you write down the directions in the wrong order, will you still make it to your friend's house? Does the order of each turn matter? Does it matter if you go left or right first?

Imagine that your friend's house and your house can be represented by points on the coordinate plane. We can use different transformations to move from one point to the next. A transformation in the coordinate plane is a change in a point's or object's location, shape, or size. There are four transformations that can take place in the coordinate plane: dilations, rotations, reflections, and translations.

Each transformation is used to alter a point or object in one specific way. A dilation changes the object's size by making it bigger or smaller. A rotation turns a point or an object a certain number of degrees around another specified point. A reflection flips a point or object over a line or another point. A translation moves a point or an object left or right and up or down.

Each transformation can be done separately or two or more can be composed together. A composition of transformations is when two or more transformations are performed on the same point or object. You can think of it as if your directions told you to drive East 5 miles and then turn left. Driving East is a translation to the right and turning left is a rotation.

Transformation Rules

Before we explore our composition examples, here are some basic transformation rules for coordinates to avoid getting lost along the way:

Dilations

If you're making an image bigger or smaller by a factor of c, simply multiply the coordinates by c: (x, y) -> (cx, cy)

Rotations

Pay attention to whether the coordinates are being rotated counter-clockwise or clockwise, since that can affect the outcome:

Counter-Clockwise Clockwise Rule
90 degrees 270 degrees (x, y) -> (-y, x)
180 degrees 180 degrees (x, y) -> (-x, -y)
270 degrees 90 degrees (x, y) -> (y, -x)

Reflections

We'll stick to reflecting over the x-axis and y-axis for now:

  • To reflect a point over the x-axis, (x, y) -> (x, -y)
  • For a point reflected over the y-axis, (x, y) -> (-x, y)

Translations

Translations are usually pretty straightforward:

Translation Rule
move left subtract from x
move right add to x
move down subtract from y
move up add to y

Example 1:

Complete a composition of transformations by rotating the original figure 90 degrees counter-clockwise, then translating the image right two units and down five units.


Original location is (3, 2)
example 2


Step 1: First, we need to rotate the original figure 90 degrees counter-clockwise. The rule for that rotation is (x, y)->(-y, x). So, our coordinates (3, 2) become (-2, 3).


black is the original and red is our rotated image
rotation


Step 2: Now, we take the new image (-2, 3) and translate it to the right 2 units and down 5 units. (x, y) -> (x +2, y - 5)


black is the original, red is our rotated image and blue is now the translation image
translation


Our final point has the coordinates (0, -2).

Example 2:

Using the same original point as in example 1, complete the composition in example 1, but in reverse order. Will your final image be the same as it was in example 1?

In this case, we are going to start with the same coordinates (3, 2), but we are first going to perform a translation of right two units and down five units. Algebraically, that looks like (3 + 2, 2 - 5) = (5, -3)


black is our original image, blue is our translation image
translation


Now, we will perform the rotation of 90 degrees counter-clockwise. We will take the coordinates after the translation, (5, -3) and use the rotation of 90 degree counter-clockwise rule (x, y)->(-y, x) so (5, -3) becomes (3, 5).


black is our original image, blue is our translation image and red is our rotation image.
rotation


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