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Name:___________________________ Date:______ Period:_____ Transformations – Easter Break Packet Ms. Anderle Transformations Line Reflection: A line reflection is a transformation in which a figure is reflected over a given line as if in a mirror. Each point of the reflection image is the same distance from a line of reflection as the corresponding point in the original figure. Basically, a line reflection “flips” an object over the line so that the image appears to be “backwards” much like how a reflected image would appear in a mirror. Under a line reflection, distance, angle measure, collinearity, and betweenness are preserved. Line Reflection Mapping Rules Reflections in the x-axis Reflections in the y-axis Reflections in the y = x line Reflection in the y = -x line Reflection in the line y = k (where k is a constant) Reflection in the line x = k (where k is a constant) rx-axis(x, y) = (x, -y) ry-axis(x, y) = (-x, y) ry = x(x, y) = (y, x) ry=-x(x, y) = (-y, -x) ry = k(x, y) = (x, 2k – y) rx = k(x, y) = (2k – x, y) These rules are applied in the following model problems. Example 1: Find a reflection in the x-axis of figure ABCD with coordinates A(-4,3), B(-3,5), C(4,6), and D(0,2). Solution: Using our mapping rule, the image points are A’(-4,-3), B’(-3,-5), C’(4,-6), and D’(0,-2). Example 2: Find the reflection across the y = x line of ΔABC with coordinates A(2,1), B(9,6), and C(5,-4). Solution: Using our mapping rule, the image points are A’(1,2), B’(6,9), and C’(-4,5). Homework Problems: 1. Find the reflection in the y = -x line of ΔPQR with the coordinates P(2,2), Q(5,3), and R(3,-1). 2. Find the reflection of the figure whose coordinates are A(-1,3), B(4,6), C(7,2) across the line represented by x = 2. 3. Find the reflection of the figure whose coordinates are A(-3,-7), B(3,-4), and C(9,-6) across the line represented by y = -3. 4. State the coordinates of the point (-2,5) under the reflection a. in the line y = x b. in the line y = 4 c. in the line x = -2 5. Find the image of the points below in reflected in the line y = -2 a. (1, 4) b. (-3, 3) c. (4, -5) Line Symmetry If a figure has line symmetry, a line can be drawn through the figure such that both sides of the figure are “mirror images” of each other. Such a line is also called the axis of symmetry. There can be more than one line of symmetry in a figure, or there may be none. One line of symmetry H Two lines of symmetry Four lines of symmetry Infinite number of lines of symmetry F No lines of symmetry Homework Problems 1) Which letter has vertical line symmetry? a. S b. N c. A d. B 2) Which letter has vertical but not horizontal line symmetry? a. X b. O c. V d. E Draw in the line symmetry to the problems below. 3) X 4) M 5) U Point Reflection and Symmetry If a figure is reflected in or through point P, then P is the midpoint of the line segement joining each point to its corresponding image. Point Reflection Mapping Rules: A reflection through the origin is written r origin(x, y) = (-x, -y) or rO(x, y) = (-x, -y), where point O is the origin. In coordinate geometry, the usual point of reflection is the origin. If point B(2, 1) is reflected through the origin, then B’(-2, -1). A reflection through a given point is written as r (h, k)(x, y) = (2h – x, 2k – y). Example: Find the image of (-4, 2) under the reflection in the point (2, 2) r(2, 2)(-4, 2) = (2·2 - -4, 2·2 – 2) = (8, 2) A figure is said to have point symmetry if the figure coincides with itself when reflected through a point or when rotated 180° about a point. The point that is the center of reflection or rotation is called the point of symmetry. F G Z SN Shapes without point symmetry , Shapes with point symmetry , , , Homework Questions 1. What letter has both point symmetry and line symmetry? a. A b. H c. E d. S 2. What is the image of (k, 2k) after a reflection through the origin. 3. Find the image of (-4, 2) under the reflection in the point (2, 2). 4. Find the image of (-2, -6) under a point reflection in the origin. 5. Find the image of (3, 2) under the reflection in the point (-1, -1) , Rotations: A rotation is a transformation in which a figure is turned around a point called the point of rotation. In the coordinate plane, this point is typically the origin. Rotations that are counterclockwise are rotations of a positive degree measure. Rotations that are clockwise are of a negative degree measure. All rotations are assumed to be clockwise about the origin unless otherwise stated Rotation Mapping Rules R90°(x, y) = (-y, x) R180°(x, y) = (-x, -y) R270˚(x, y) = (y, -x) R360°(x, y) = (x, y) Rotation of 90° and -270° Rotation of 180° and -180° Rotation of 270° and -90° Rotation of 360° and -360° Homework Questions 1. What is the image of (-2, 3) under a rotation of: a) 90° b) 180° c) 270° d) -90° e) -270° 2. Name the coordinates of each point under a rotation of 90° a) (5,1) b) (-3,3) c) (8, -2) d) (0,5) 3. Name the coordinates of each point under a rotation of 270° a) (-3,2) b) (-2,-2) c) (4,4) d) (7,14) Translation A translation is a transformation in which each point in a figure slides a certain distance and in the same direction. On the coordinate plane, the translation is defined by the mapping rule: Ta, b(x, y) = (x + a, y + b). Model Problems: 1. What is the image of point P(-3, 2) under the transformation T-2, 6? Answer: T-2, 6 means add -2 to the x-value (-3) and add 6 to the y-value (2). Therefore, the image of point P is (-5, 8) 2. A transformation maps P(x,y) onto P’(x – 4, y + 2). Under the same transformation, what are the coordinates of Q’, the image of Q(2, -3)? Answer: Since the given transformation is T-4, 2, Q(2, -3) is mapped onto Q’ by adding -4 to the x-value (2) and 2 to the y-value (-3). Q’(-2, -1). Homework Problems 1. What is the image H(-1, 3) under the translation T2, -1? 2. Find the image of (-1,5) if translation T is defined by (x, y) (x + 3, y – 1) 3. What is the image of P(-2,3) under the translation T4, -2? 4. ***If the transformation Tx, y maps point M(1,-3) onto point M’(-5,8), what is the value of x? Dilations A dilation is a transformation in which the size of a figure is changed and the figure is moved. The dilation mapping rule can be expressed as Dk(x, y) = (kx, ky). Model Problems: 1. D3(x, y) = (3x, 3y) 2. D1/2(2, 4) = (1, 2) Note: -If k = 1, then the image is identical to the original. -If k = -1, then the image is congruent and is the same as a point reflection. -If |k| > 1, then the image is similar and larger than the original. -If |k| < 1, then the image is similar and smaller than the original. Homework Problems 1) What are the coordinates of point (2, -4) under the dilation D-2? 2) What are the coordinates of point (9, -6) under the dilation D1/3? 3) What are the coordinates of point (12, -4) under the dilation D5? 4) If the dilation Dk(2, -4) = (-1, 2), the scale factor k is equal to: 5) Find the image of (6, -9) under the dilation D2/3.