CCSSM Stage 3 Companion Text

1. Lesson 3-P CCSSM Stage 3 Companion Text Composition of Transformations

2. Warm-Up Write the transformation rule for each transformation. 1. A reflection over the x-axis. 2. A dilation with a scale factor of 5. 3. A translation down 3 units. 4. A translation right 1 unit and up 5 units.

3. Lesson 3-P Composition of Transformations Target: Perform multiple transformations on a figure including a combination of reflections, rotations, translations and dilations.

4. Explore! On a Trip Step 1 Graph the point A(3, 3) on a coordinate plane. Step 2Let point B represent point A under the transformation . Graph B and give the ordered pair for its location. Step 3Let point C be a reflection of point B over the y-axis. Graph point C and give its ordered pair. Step 4 Let point D be a rotation of point C 90o clockwise about the origin. Graph point D and give its ordered pair. Step 5 Let point E represent point D under the transformation . Graph point E and give its ordered pair. Step 6 Write a translation rule that moves point E back to the original location (point A). Step 7 Create your own “trip” around the coordinate plane for someone else to follow. Start by giving the ordered pair for a point you choose to be point S. Give four transformation directions that will move the points to four different locations, STUVW, around the plane. Create an answer key for your path by giving an ordered pair for points T, U, V and W.

5. Vocabulary Composition of Transformations A series of transformations. Good to Know!  The transformations that are included in the sequence will determine whether or not the image is similar or congruent to the original figure.  If a dilation occurs on a figure during a transformation sequence, the image will be similar to the pre-image.  Any combination of translations, reflections and rotations will create an image that is congruent to the original figure.  On a coordinate plane, you can prove two figures are similar or congruent if you can write a sequence of transformations that maps one figure onto the other.

6. Example 1 Use the triangle formed by the points H(−2, 4), K(1,2) and J(0, −3). a. Graph HꞌKꞌJꞌ by reflecting HKJ over the x-axis and then following the translation rule (x, y)  (x – 4, y + 3). b. Are the figures congruent or similar to each other? How do you know? Graph the original figure, HKJ H K J

7. Example 1 Continued… Use the triangle formed by the points H(−2, 4), K(1,2) and J(0, −3). a. Graph HꞌKꞌJꞌ by reflecting HKJ over the x-axis and then following the translation rule (x, y)  (x – 4, y + 3). Reflection over the x-axis is . Use the transformation rule to identify the coordinates of HKJ after it is reflected. H K J

8. Example 1 Continued… Use the triangle formed by the points H(−2, 4), K(1,2) and J(0, −3). a. Graph HꞌKꞌJꞌ by reflecting HKJ over the x-axis and then following the translation rule (x, y)  (x – 4, y + 3). Translation 4 left and up 3. Use the transformation rule to identify the coordinates of HKJ after it is translated. Jꞌ H K Kꞌ Hꞌ J

9. Example 1 Continued… Use the triangle formed by the points H(−2, 4), K(1,2) and J(0, −3). b. Are the figures congruent or similar to each other? How do you know? The figures are congruent because HꞌKꞌJꞌ is a result of two transformations (reflection and translation) on HKJ. This series of transformations creates congruent images. Jꞌ H Jꞌ K Hꞌ J

10. Example 2 Write a series of two transformations that shows MNP is similar to MꞌNꞌPꞌ. The triangle appears to be enlarged and shifted to the right. This can be done using a dilation and a translation. Record the original coordinates. M(0, 0), N(4, 0), P(0, 3) Pꞌ Different combinations of transformations in different orders could create this movement, but one of them must be a dilation. P M Mꞌ N Nꞌ

11. Example 2 Continued… Write a series of two transformations that shows MNP is similar to MꞌNꞌPꞌ. The image appears to be larger. Find the scale factor using corresponding sides of MNP and MꞌNꞌPꞌ. Multiply each vertex of MNP by a scale factor of 2. Pꞌ P M Mꞌ N Nꞌ

12. Example 2 Continued… Write a series of two transformations that shows MNP is similar to MꞌNꞌPꞌ. The point at the origin is shifted 2 units to the right. Add 2 units to each x-coordinate for a translation rule of . Mꞌ(2, 0), Nꞌ(10, 0), Pꞌ(2, 6) Since these points match MꞌNꞌPꞌ on the graph above, the series of transformations maps MNP onto MꞌNꞌPꞌ. dilation translation Pꞌ P M Mꞌ N Nꞌ

13. Example 2 Continued… Write a series of two transformations that shows MNP is similar to MꞌNꞌPꞌ. The two figures are similar (not congruent) because a dilation was part of the transformation sequence that made the triangle larger. Pꞌ P M Mꞌ N Nꞌ

14. Exit Problems Use the point T(4, 3). Write the ordered pair for the final location of the given point after completing the transformations in the order listed. 1. Reflection over the x-axis; (x,y) → (x + 2, y − 4) 2. Dilation with a scale factor of 3; (x,y) → (x + 1, y) 3. Reflection over the y-axis; (x, y) → (x − 6, y + 3) 4. Reflection over the y-axis; Reflection over the x-axis

15. Communication Prompt How many different compositions of transformations can map A(8, 1) onto A'(4, 2)? Explain your reasoning.