Consider the following definition:[br][br]Definition 4.1: Transformations are functions from the plane to the plane.[br][br]Below is a series of demonstrations showing various transformations. Each transformation maps one figure to another in the plane. If a transformation were to map a figure out of the plane into, for example, 3-space, it would no longer satisfy our definition, and would therefore no longer be a transformation. The transformed figure must remain in the plane.[br]The first transformation we will examine is called a translation. In the applet below, drag the different parts of the translation vector around in the applet window, and record your observations. Use your observations to answer the questions below.

1. What do you notice about Translations?[br][br]2. Why do you think one figure is called the pre-image and the other is called the image? What is the difference between the two figures?[br][br]3. Imagine you need to explain what a translation is to a friend. Using a different polygonal figure (not a triangle), write down an explanation of what you think a translation is.[br][br][br]The second transformation we will examine is a rotation. In the applet window below, drag each point in the applet (don't forget the point of rotation) and observe what happens. Try pausing the animation. Do you notice anything new when the animation is static? Try adjusting the angle manually. use your observations to answer the questions below.

4. What do you notice about rotations?[br][br]5. Does anything change when the point of rotation is inside the pre-image instead of outside?[br][br]6. What changes and what stays the same?[br][br]7. Just like above, write how you would explain to your friend what you think a rotation is using the same polygon that you used from question 3 above.[br][br][br]The applet below shows a translation and rotation together. Do you notice anything that you missed before? Try playing with the applet to gather more observations. Use your observations to answer the questions below.

7. In the applet, the point of rotation is fixed in the center of the figure. Does it have to be in the center?[br]8. The above animation combines two transformations. Can more than two transformations be combined?[br]9. Suppose you were to translate, rotate, and then translate a figure. Can produce the same figure with only two transformations?[br]10. Does this applet change how you would explain these transformations to your friend?[br][br]The third transformation we will examine is a reflection. Play with the applet below and record your observations.

11. What do you notice about reflections?[br]12. Is a 180 degree rotation around a point the same as a reflection about a line? Why or why not?[br][br][br]Play with the Double Reflection applet below, and use your observations to answer the questions.

13. How are reflections and rotations related?[br]14. How many reflections are required for a rotation? Can 3 reflections be a rotation? 4 rotations? 5 rotations? Any number of rotations? Describe the pattern.[br][br]Examine below the reflection about a point, and play with the figures in the applet. What do you notice? Use your observations to answer the questions below.

15. What do you notice? Is reflection about a point the same as reflection about a line? Why or why not?[br]16. Is reflection about a point the same as a 180 degree rotation? Why or why not?[br]17. Using the same polygon as before, explain reflections to your friend. Make sure to explain reflections about a line and a point, how they are different from each other, and how they are related to rotations.[br][br][br]The fourth and final transformation we will examine is a dilation. Play with the applet and use your observations to answer the questions below. Pay special attention to what makes dilation unique from the other transformations.

18. When the pre-image is transformed to the image, what changes and what stays the same?[br]19. What changes in a dilation that doesn't change in any of the previous transformations?[br]20. What stays the same in a dilation that also stays the same in the previous transformations?

At the very beginning, we said that transformations are functions. A translation is a function that translates each individual point of a geometric figure in the plane. A coordinate grid can be applied to the geometric figure. Then each point (x,y) can be translated to (x-a, y-b) with the parameters a and b. For a rotation of angle t about a point (a,b), the point (x,y) is rotated by (xcos(t) - ysin(t) - a, xsin(t) + ycos(t) - b) with the parameters a, b, and t. It is important at this point to make a distinction between parameters and variables. Both are represented by variables, but they serve a different purpose. The letters a, b, and t are parameters. They are constants chosen for the function based on the desired result. For a translation, a and b are the distances we want each x and y coordinate to be translated, while x and y are the variables of the function. For a rotation, a and b are the fixed coordinates of the center of rotation, and t is the fixed angle of rotation. In both, x and y are the function variables. The variables are what create the figure, or draw the graph so to speak. Consider the rotated parabola below xsin(t) + ycos(t) - b = (xcos(t) - ysin(t) - a)^2.

Pause the animation above. Notice that the three parameters uniquely determine a transformation. Every time the parameters are changed, a new transformation is produced. It is easy to mistake the series of transformations created by the animation as a single moving transformation. It is actually many transformations shown rapidly one after the other.

Definition 4.2: An isometry is a transformation that preserves lengths and angle lkjdfmeasures.Consider the following definition:[br]Definition 4.2: An isometry is a transformation that preserves lengths and angle [br]measures.[br][br]You may have noticed that translations, rotations, and reflections all preserve both segment lengths and angles in the transformed polygons. These transformations are all called isometries. However, Dilations are not isometries. Can you see why?[br][br]Recall from chapters 1 and 2:[br]Postulate 1.9: Segment Length and Congruence Postulate.Two line segments are Postulate 1.9: Two line segments are congruent if and only if their lengths are equal.[br]congruent if and only iftheir lengths are equal. Postulate 2.3: Two angles are congruent if and only if their angle measures are equal.[br][br]For Polygons, we use the word congruent with similar meaning. Informally, you can think of two polygonal figures as congruent if their corresponding angle measures and segment lengths are congruent. We will use a theorem to formally define congruent figures.

Theorem 4.1: Two figures in the plane are congruent if and only if a sequence of isometries maps one figure onto the other.[br][br]Since we are restricted to isometries, we are not allowed to use dilation. You have already seen this theorem in action in the demonstrations above. Above is a sequence of isometric transformations. You may notice that the figures from the first and second transformation are both the image of the previous figure and the pre-image of the following figures. Note also that the final figure could be chosen as the beginning figure, and it would map to the original figure by reversing the steps.[br][br]Below is a sequence of transformations that demonstrate similar figures.[br][br]Theorem 4.2: Two figures in the plane are similar if and only if a sequence of [br]translations, rotations, reflections, or dilations maps one figure onto the other.

Below is one final transformation: reflection about a circle. Are the images congruent, similar, or neither, to the pre-image. Why or why not? Play with the applet to explore the question.

Earlier we found that some transformations or series of transformations are equivalent to others. Find another example not previously explored of a series of transformations that is equivalent to a single transformation.[br]Consider the reflection about a circle. Can this transformation be used in a series of transformations to produce either congruent or similar figures? Why or why not?[br]Give an example of a series of transformations that produces congruent figures, and an example for similar figures.[br]Explain how variables and parameters are different.[br]Consider how domain and pre-image are similar, as well as range and image. Why do we use different terminology? In the example of the rotated and translated parabola, does it have a domain and range AND a pre-image and image, or just one or the other?[br]Find a friend who doesn't know what geometric transformations are. Using the polygon you chose at the beginning (not a triangle), explain what geometric transformations are with your own examples. Write a short reflection on your lesson with your friend.