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 : Defining Rotations, Reflections, and Translations

Key Concepts The coordinate plane is separated into four quadrants, or sections:  In Quadrant I, x and y are positive.  In Quadrant II, x is negative and y is positive.  In Quadrant III, x and y are negative.  In Quadrant IV, x is positive and y is negative : Defining Rotations, Reflections, and Translations

Key Concepts, continued A translation is an isometry where all points in the preimage are moved parallel to a given line. No matter which direction or distance the translation moves the preimage, the image will have the same orientation as the preimage. Because the orientation does not change, a translation is also called a slide. Translations are described in the coordinate plane by the distance each point is moved with respect to the x- axis and y-axis. If we assign h to be the change in x and k to be the change in y, we can define the translation function T such that T h, k (x, y) = (x + h, y + k) : Defining Rotations, Reflections, and Translations NOTE: I ADDED THIS BOTTOM BULLET TO FILL PAGE A BIT MORE. IN THE BOOK, IT GOES AFTER THE NEXT. IF THIS ISN’T OK, WE CAN FIX.

Key Concepts, continued In the translation below, we can see the points A, B, and C are translated along parallel lines to the points,, and. Each point is carried the same distance and direction, so not only is congruent to, it also maintains the same orientation : Defining Rotations, Reflections, and Translations

Key Concepts, continued A reflection is an isometry in which a figure is moved along a line perpendicular to a given line called the line of reflection. Unlike a translation, each point in the figure will move a distance determined by its distance to the line of reflection. In fact, each point in the preimage will move twice the distance from the line of reflection along a line that is perpendicular to the line of reflection. The result of a reflection is the mirror image of the original figure; therefore, a reflection is also called a flip : Defining Rotations, Reflections, and Translations

Key Concepts, continued As seen in the image on the previous slide, A, B, and C are reflected along lines that are perpendicular to the line of reflection,. Also note that the line segments,, and are all of different lengths. Depending on the line of reflection in the coordinate plane, reflections can be complicated to describe as a function. Therefore, in this lesson we will consider the following three reflections. through the x-axis: r x-axis (x, y) = (x, –y) through the y-axis: r y-axis (x, y) = (–x, y) through the line y = x: r y = x (x, y) = (y, x) : Defining Rotations, Reflections, and Translations

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 : Defining Rotations, Reflections, and Transformations

Key Concepts, continued 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. A figure may be rotated clockwise, in the direction that the hands on a clock move, or counterclockwise, in the opposite direction that the hands on a clock move : Defining Rotations, Reflections, and Translations

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 : Defining Rotations, Reflections, and Translations

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: R 90 (x, y) = (–y, x) 180° rotation about the origin: R 180 (x, y) = (–x, –y) 270° rotation about the origin: R 270 (x, y) = (y, –x) : Defining Rotations, Reflections, and Translations

Common Errors/Misconceptions confusing reflections and rotations (For example, a 90° rotation from Quadrant I to Quadrant II may appear similar to a reflection through the y-axis.) finding that the final output of combinations of rotations and reflections has the same outcome as a single transformation : Defining Rotations, Reflections, and Translations

Guided Practice Example 1 How far and in what direction does the point P (x, y) move when translated by the function T 24, 10 ? : Defining Rotations, Reflections, and Translations

Guided Practice: Example 1, continued 1.Each point translated by T 24,10 will be moved right 24 units, parallel to the x-axis : Defining Rotations, Reflections, and Translations

Guided Practice: Example 1, continued 2.The point will then be moved up 10 units, parallel to the y-axis : Defining Rotations, Reflections, and Translations

Guided Practice: Example 1, continued 3.Therefore, T 24,10 (P) = = (x + 24, y + 10) : Defining Rotations, Reflections, and Translations ✔

Guided Practice: Example 1, continued : Defining Rotations, Reflections, and Translations

Guided Practice Example 2 Using the definitions described earlier, write the translation T 5, 3 of the rotation R 180 in terms of a function F on (x, y) : Defining Rotations, Reflections, and Translations

Guided Practice: Example 2, continued 1.Write the problem symbolically. F = T 5, 3 (R 180 (x, y)) : Defining Rotations, Reflections, and Translations

Guided Practice: Example 2, continued 2.Start from the inside and work outward. R 180 (x, y) = (–x, –y) Therefore, T 5, 3 (R 180 (x, y)) = T 5, 3 (–x, –y) : Defining Rotations, Reflections, and Translations

Guided Practice: Example 2, continued 3.Now translate the point. T 5, 3 (–x, –y) = (–x + 5, –y + 3) : Defining Rotations, Reflections, and Translations

Guided Practice: Example 2, continued 4.Write the result of both translations. F = T 5, 3 (R 180 (x, y)) = (–x + 5, –y + 3) : Defining Rotations, Reflections, and Translations ✔

Guided Practice: Example 2, continued : Defining Rotations, Reflections, and Translations