Introduction

1. Introduction Now that we have built our understanding of parallel and perpendicular lines, circular arcs, functions, and symmetry, we can define three fundamental transformations: translations, reflections, and rotations. We will be able to define the movement of each transformation in the coordinate plane with functions that have preimage coordinates for input and image coordinates as output. Day 3 Defining Rotations,

2. Key Concepts The coordinate plane is separated into four quadrants, or sections: In Quadrant I, xand y are positive. In Quadrant II, xis negative and y is positive. In Quadrant III, xand y are negative. In Quadrant IV, xis positive and y is negative. ( -,+) ( +,+) ( +,-) ( -,-) Day 3 Defining Rotations,

3. Key Concepts, continued A rotation is an isometry where all points in the preimage are moved along circular arcs determined by the center of rotation and the angle of rotation. A rotation may also be called a turn. This transformation can be more complex than a translation or reflection because the image is determined by circular arcs instead of parallel or perpendicular lines. Similar to a reflection, a rotation will not move a set of points a uniform distance. When a rotation is applied to a figure, each point in the figure will move a distance determined by its distance from the point of rotation. Day 3 Defining Rotations,

4. Key Concepts, continued The figure below shows a 90° counterclockwise rotation around the point R. Comparing the arc lengths in the figure, we see that point B moves farther than points A and C. This is because point B is farther from the center of rotation, R. 5.2.1: Defining Rotations, Reflections, and Translations

5. Let’s see how a 90˚rotation changes the x and y-values. Rotate Point A 90˚ Counterclockwise Around the origin A’ A a)Write the ordered pair for A _______________ (4,3) b) Write the ordered pair for A’ _______________ (-3,4) c) How were the values of x and y affected? ______________

6. Now Let’s see how a 180˚rotation changes the x and y-values. Rotate Point A 180˚ Counterclockwise Around the origin A a)Write the ordered pair for A _______________ (4,3) b) Write the ordered pair for A’ _______________ (-4,-3) A’ c) How were the values of x and y affected? ______________

7. Now Let’s see how a 270˚rotation changes the x and y-values. Rotate Point A 270˚ Counterclockwise Around the origin A a)Write the ordered pair for A _______________ (4,3) b) Write the ordered pair for A’ _______________ (3,-4) A’ c) How were the values of x and y affected? ______________

8. Key Concepts, continued Depending on the point and angle of rotation, the function describing a rotation can be complex. Thus, we will consider the following counterclockwise rotations, which can be easily defined. 90° rotation about the origin: R90(x, y) = (–y, x) 180° rotation about the origin: R180(x, y) = (–x, –y) 270° rotation about the origin: R270(x, y) = (y, –x) Day 3 Defining Rotations,

9. Example 1 A triangle is formed by the points A(2, 6), B(3, 3), and C(6, 5). Give the ordered pairs of the image R90 .

10. Example 2 A triangle is formed by the points A(2, 6), B(3, 3), and C(6, 5). Give the ordered pairs of the image R180 .

11. Example 3 A triangle is formed by the points A(2, 6), B(3, 3), and C(6, 5). Give the ordered pairs of the image R270 .

12. Example 4 Given the points (3,-2), what is T 7,-2? Example 5 If a triangle is formed by the points A(2,6), B(5,-3) C(2,-5), give the ordered pairs of the image after a transformation of rx-axis. Example 6 If a triangle is formed by the points A(2,6), B(5,-3) C(2,-5), and the ordered pairs after a transformation are A(6,2), B(-3,5), C(-5,2).