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Geometry Name________________________ Transformations (3): Transformation Equations 3/26/07 Transformations of the plane A transformation of the plane is a function whose domain and range are the xy-plane (or sets of points within the plane). You are already familiar with geometric descriptions of certain kinds of transformations, such as translations, reflections, rotations, and dilations. A pair of transformation equations defines the mapping of the coordinates (x,y) to the coordinates of the image (x’,y’). Here is an example of a pair of transformation equations: x = x + 3 y y – 1 = These equations say that given any point (x, y), the image point ( x y is found by adding 3 to the , ) x-coordinate and subtracting 1 from the y-coordinate. In other words, these equations define a horizontal translation by 3 units to the right and a vertical translation by 1 unit down. Many geometrically interesting transformations have fairly simple equations. In fact, all translations, reflections, rotations, and dilations can be described using equations of the form x = ax + by + e y cx + dy + f = An important fact about transformation equations having this form is that the image of a line segment is always a line segment. Therefore, if you want to see the effect of a transformation on a polygon, you just need to find the images of the vertices. Problems 1. Give a geometric description of the transformation defined by each pair of equations. Be as specific as possible. (For example, for a reflection, identify the reflection line.) If you’re not sure about an answer, try graphing a few pairs of input and output points. a. x = x – 2 f. x = 6 – x y = y + 5 y y = b. x = 2x g. x = y y = y y = x c. x = x h. x = –y y 1 y = 3 y = x d. x = x i. x = x y = –y y = 0 e. x = –x j. x = kx y = –y y = y + c Geometry Transformations (3) 2. Write a pair of transformation equations for each of the following. If you’re not sure, write down the coordinates for a few pairs of input and output points, try to spot a pattern, then turn the pattern into general rules for finding x and y . a. reflection across the y-axis b. the translation that maps the point (3, –1) to an image of (8, –4) c. a horizontal dilation that would shrink any object to half of its original width d. reflection across the horizontal line y = 4 e. reflection across the line y = –x f. rotation by 180° centered at the origin g. rotation clockwise by 90° centered at the origin h. projection onto the y-axis (that is, a transformation that maps every point to the point on the y-axis having the same y-coordinate) 3. The transformation x = 4x + 2y, y = 2x + y maps every point of the plane onto a line L. By experimenting with different points and their images, find the equation of line L. 4. Consider the transformation: x = ax + by + e y = cx + dy + f a. Calculate the images of the vertices of the unit square with vertices: (0, 0), (0, 1), (1, 0), and (1, 1). b. What special type of quadrilateral is the image of the unit square? Prove your answer. (For example: if you think it’s a rectangle, prove the angles are right angles; if you think it’s a parallelogram, prove the sides are parallel; etc.) 5. A transformation having the form x = ax + by (note there is no e term) y = cx + dy (note there is no f term) is called a linear transformation. a. Which of the transformations from problem 2 are linear transformations? b. Which of the transformations from problem 3 are linear transformations? c. Of the four well-known types of transformations (translations, reflections, rotations, dilations), there’s one type that’s almost never a linear transformation. Which type?

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