Introduction The word transform means “to change.” In geometry, a transformation changes the position, shape, or size of a figure on a coordinate plane. The original figure, called a preimage, is changed or moved, and the resulting figure is called an image. We will be focusing on three different transformations: translations, reflections, and rotations. These transformations are all examples of isometry, meaning the new image is congruent to the preimage. 1

In this lesson, we will learn to describe transformations as functions on points in the coordinate plane. The potential inputs for a transformation function f in the coordinate plane will be a real number coordinate pair, (x, y), and each output will be a real number coordinate pair, f (x, y) the x and y values will change. Example: f (x, y) = (x + 1, y + 2) means for any ordered pair (x, y) add 1 to the x coordinate and add 2 to the y coordinate 2

Finally, transformations are generally applied to a set of points such as a line, triangle, square or other figure. In geometry, these figures are described by points, P, rather than coordinates (x, y), and transformation functions are often given the letters R, S, or T We will see T(x, y) written T(P) or P', known as “P prime.” A transformation T on a point P is a function where T(P) is P'. 3

Example 1 Given the point P(5, 3) and T(x, y) = (x + 2, y + 2), what are the coordinates of T(P)? 4

Guided Practice: Example 1, continued 1.Identify the point given. We are given P(5, 3) : Transformations As Functions

Guided Practice: Example 1, continued 2.Identify the transformation. We are given T(P) = (x + 2, y + 2) : Transformations As Functions

Guided Practice: Example 1, continued 3.Calculate the new coordinate. T(P) = (x + 2, y + 2) (5 + 2, 3 + 2) (7, 5) T(P) = (7, 5) : Transformations As Functions ✔

We can also define transformations using a subscript notation. T(x, y)=(x + 2, y - 1) can be defined as T 2, -1 meaning add two to the x value and subtract one from the y value. 8 Plot the points A’, B’, C’ using the translation T 2, -1 A B C A’ B’ C’ Example 2 A(1, 2) => B(4, 4) => C(3, 0) => A’(3, 1) B’(6, 3) C’(5, -1)

Guided Practice Example 3 Given the transformation of a translation T 5, –3, and the points P (–2, 1) and Q (4, 1), show that the transformation of a translation is isometric by calculating the distances, or lengths, of and : Transformations As Functions

Guided Practice: Example 3, continued 1.Plot the points of the preimage : Transformations As Functions

Guided Practice: Example 3, continued 2.Transform the points. T 5, –3 (x, y) = (x + 5, y – 3) : Transformations As Functions

Guided Practice: Example 3, continued 3.Plot the image points : Transformations As Functions

Guided Practice: Example 3, continued 4.Calculate the distance, d, of each segment from the preimage and the image and compare them. Since the line segments are horizontal, count the number of units the segment spans to determine the distance. d(PQ) = 5 The distances of the segments are the same. The translation of the segment is isometric : Transformations As Functions ✔

Guided Practice: Example 3, continued : Transformations As Functions