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Transformations

Transformations. Unit 7. Section 1. Introduction to Transformations & Translations. What is happening in these pictures?. Insert picture of translation and rotation here. Do Now. Take out a sheet of paper and answer the following questions: What areas did I do well on in the test?

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Transformations

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  1. Transformations Unit 7

  2. Section 1 Introduction to Transformations & Translations

  3. What is happening in these pictures? • Insert picture of translation and rotation here

  4. Do Now • Take out a sheet of paper and answer the following questions: • What areas did I do well on in the test? • What areas do I still need to work on? • What can I do to improve/keep up the good work for the next test?

  5. What is happening in these pictures? • Insert picture of reflection and dilation here

  6. Transformations • Transformation—the process of moving a figure or changing it • Map—to move • The location of the figure after it is transformed is called the transformation image • Example: The image of point A after it is mapped through a translation is point A’.

  7. Four Main Types of Transformations • Translation • Rotation • Reflection • Dilation

  8. Translation • Animation 1 • What is happening in the animation?

  9. Definition of Translation • Moving a figure to a new location with no other changes • Remember: • Translate = Slide • Every point moves the same number of units in the same direction

  10. Notation • Transformations: add apostrophe to letter for each point: • Original Object (A)  Transformation Image (A’) • Example: Translation of triangle ABC is triangle A’B’C’

  11. Translations on the Coordinate Plane Continued • (x + h, y + k) • Positive = shift up or to the right • Negative = shift down or to the left • Also can be written as <h, k> • Ex.: Point A(-1, -5) transforms according to the rule (x, y)  (x + 2, y + 3). What is the resulting point? • Answer: A’(-1 + 2, -5 + 3) = A’(1, -2)—two units right, three units up

  12. Practice Problems I • State the transformation image (result) after each transformation. Describe what is happening in words. • (-1, 3)  (x – 5, y + 3)  • (5, -12)  (x – 7, y + 1)  (-6, 6) (-2, -11)

  13. Practice Problems II • A triangle with coordinates L(-3, 2), M(6, -2), and N(2, -5) is translated so that M’ is at (2, 2). Where are the images of points L and N located? • From M to M’ is (x – 4, y + 4) • L’(-3 – 4, 2 + 4) = L’(-7, 6) • N’(2 – 4, -5 + 4) = N’(-2, -1)

  14. Section 2 Rotations

  15. How are these motions different?

  16. Rotation • A rotation is a transformation where the figure is turned about a given point. • The center of rotation is the point around which the object is rotated.

  17. Rotation in the Coordinate Plane • Instead of left/right, we say clockwise (cw) or counterclockwise (ccw) • Animation 2 • A clockwise rotation of 90 degrees is equal to a counterclockwise rotation of how many degrees? • Answer: 270 degrees

  18. How to Rotate a Shape a Graph • Question: Graph the rotation of triangle ABC with points A(2, 1); B(5, 2); C(3, 4) 90 degrees ccw. • Plot the axes and points on your graph. • Rotate your paper 90° clockwise. • If the question says cw, rotate your paper ccw. • Plot the same coordinates, using the y-axis as the x-axis and the x-axis as the y-axis. • What do you notice about the coordinates?

  19. Practice Activity • Graph the image of triangle ABC with points A(2, 1); B(5, 2); C(3, 4) under a rotation of… • 180° ccw • 270° ccw • What do you notice about the coordinates?

  20. Special Rotations • 90° ccw (270° cw)—image has coordinates (-y, x); (2, 1)  (-1, 2) • 180° ccw or cw—image has coordinates (-x, -y); (2, 1)  (-2, -1) • 270° ccw (90° cw)—image has coordinates (y, -x); (2, 1)  (1, -2)

  21. Practice Problems • Find the coordinates of these points when rotated 90° ccw, 180° ccw, and 270° ccw : (5, -3) (3, 5) (-5, 3) (-1, 2) (-2, -1) (1, -2) (4, 0) (0, 4) (-4, 0)

  22. Section 3 Reflections

  23. Reflection • A transformation where each point appears at an equal distance on the opposite side of a given line • Line of reflection • Animation 1

  24. What do you notice? • What parts of the shape stay the same? • Which part changes? • Orientation is reversed—the object is no longer facing the same way

  25. Graphing a Reflection on the Coordinate Plane • Question: Graph the reflection of triangle ABC with points A(-2, 1); B(-5, 2); and C(-3, 4) across the x-axis. • Graph the axes and plot the points. • Fold your paper along the line of reflection. • Trace the triangle on the back of your paper. Unfold the paper and trace the triangle again, this time on the front of the paper. • Label the new coordinates.

  26. On Your Own… • Graph the reflection of triangle ABC with points A(-2, 1); B(-5, 2); and C(-3, 4): • Across the y-axis • Across the line y = x (Hint: Graph the line first!)

  27. What are the coordinates of the transformation image? • Find the coordinates of the image after a reflection across x-axis, y-axis, and line y = x. A’(-2, -1) B’(-5, -2) C’(-3, -4) A’’(2, 1) B’’(5, 2) C’’(3, 4) A’’’(1, -2) B’’’(2, -5) C’’’(4, -3)

  28. Special Reflections • Across the x-axis: (x, y)  (x, -y) • Example: (-2, 1)  (-2, -1) • Across the y-axis: (x, y)  (-x, y) • Example: (-2, 1)  (2, 1) • Across the line y = x: (x, y)  (y, x) • Example: (-2, 1)  (1, -2)

  29. Section 4 Dilations

  30. What is happening in this picture?

  31. Dilation • Transformation in which a polygon is enlarged or reduced by a factor around a center point. • “Zoom in” • “Zoom out”

  32. What kind of shapes do dilations create? • SIMILAR • “Same shape, different sizes” • Congruent angles • Proportional sides

  33. Isometry • Transformation that results in a congruent image • Examples of isometry: • Translation • Rotation • Reflection • NOT an example of isometry • Dilation

  34. Different Kinds of Isometry • Direct isometry—orientation stays the same • Translation • Rotation • Opposite isometry—orientation is reversed • Reflection

  35. Dilation in the Coordinate Plane • To find the dilation image, multiply every point by the scale factor • Scale Factor—how big the image is compared to the original • AKA. Dilation ratio • Example: Point A(-3, 2) undergoes a dilation of 2. What are the coordinate of the image? • Answer: (-3 × 2, 2 × 2) = (-6, 4)

  36. Practice Problems A’’(-6, -10) A’’’(-9, -15) A’(-1.5, -2.5) B(-7, 5) B’(-3.5, 2.5) B’’’(-21, 15) C(0, -4) C’(0, -2) C’’(0, -8) D(1, 2) D’’(2, 4) D’’’(3, 6)

  37. Section 5 Compositions

  38. Composition • Combination of two or more transformations • Examples: • Dilation and translation • Reflection and rotation

  39. What happens when we combine two or more transformations? • What is happening in this picture? • Glide reflection = reflection + translation

  40. Notation • D2 – Dilation of 2 • R90 – Rotation of 90° counterclockwise • Assume the direction is ccw unless it says “clockwise” • ry-axis – Reflect across the y-axis • T-2,3 – Translation of (x – 2, y + 3)

  41. How to Do Composition Problems • Treat composition as two problems • DO SECOND PART FIRST! • Example: A point located at A(-3, -2) undergoes the following composition: D2◦ R90. State the image of points A’. • R90: Use formula (-y, x). • A(-3, -2)  A’(2, 3) • D2 : A(4, 6

  42. Does order matter in a composition? • Try doing these two compositions. What do you notice about the answer? • Point A(-2, -4) dilated D2 ◦ T-2,3 . • A’(-8, -2) • Point A (-2, -4) dilated T-2,3 ◦D2. • A’(-6, -5) • ORDER MATTERS IN DILATION!

  43. Practice Problems • Find the image after each composition.

  44. Section 6 Coordinate Proofs with Transformations

  45. Comparing Different Transformations • Which properties are preserved under each transformation? Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes Yes Yes No Yes

  46. Parallelism • “Is parallelism preserved?” • Means: “Are parallel lines in an object still parallel after the object is transformed?” • Slopes of parallel lines are equal. • Use the slope formula to prove this:

  47. Example 1 • Given: quadrilateral ABCD with vertices A(-1,1), B(4,-2), C(3,-5), and D(-2,-2). • State the coordinates of A'B'C'D', the image of quadrilateral ABCD under a dilation of factor 2. • Prove that A'B'C'D' is a parallelogram. • Dilation of 2: • A’(-2, 2); B’(8, -4); C’(6, -10); D’(-4, -4)

  48. Example 1 Continued • Prove that A'B'C'D' is a parallelogram. • Opposite sides of a parallelogram are parallel. • Find the slopes of A’B’ and D’C’ to show that they’re equal. • Find the slopes of B’C’ and A’D’ to show that they’re equal.

  49. Example 2 C • The vertices of triangle ABC are A(2,1), B(7,2), and C(3,5). • Identify and graph a transformation of triangle ABC such that its image results in AB||A’B’. B A

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