1. Matrices for Rotations Sec. 4-8 LEQ: How can you use matrix multiplication to graph figures and their rotation images?

2. Rotations ► Magnitude: the amount and direction of the turn  Positive magnitude is a counterclockwise turn  Negative magnitude is a clockwise turn ► Rotation of magnitude x around the origin is denoted R x.  For example: R 120 R -75 R -75 120 ˚ 75 ˚

3. Composite of Two Rotations ► Rotations often occur one after the other…going from one frame to another in animated cartoons  Pg. 246 ► Theorem: A rotation of a ˚ following a rotation of b ˚ with the same center results in a rotation of (a + b) ˚.  In symbols: R a ◦ R b = R a+b

4. Matrices for Rotations ► Recall: r y=x ◦ r x = R 90 ► is the matrix for R 90 ► By composing two 90 ˚ rotation, a matrix for R 180 can be found  R 90 ◦ R 90 = R 180  So, is the matrix for R 180

5. Your Turn… ► Find the matrix for R 270.  R 180 ◦ R 90 = R 270  So, is the matrix for R 270.

6. Negative Magnitudes ► The image of any figure under a rotation with a negative magnitude can be found by a clockwise rotation.  For example: R -90 represents a 90 ˚ turn clockwise. Because a rotation of -90 ˚ has the same images for every point as a rotation of 270 ˚, R -90 equals R 270.  And, R -180 = R 180  And, R -270 = R 90

7. For example ► Triangle ABC has coordinate A = (0, 4), B = (4, 2) and C = (1, -2). Find the coordinates of the image of triangle ABC under R -90.  R -90 = R 270  Therefore,  So, A’ = (4, 0), B’ = (2, -4) and C’ = (-2, -1)

8. Your Turn… ► Lesson Master 4-8A #5 & 6

9. Homework ► Pgs. 248-250 #1-8, 10-14, 16-23, 28