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Chapter 9

Chapter 9. Transformations. 9.1 Reflections. Types. There are four types of transformations: Reflections Translations Rotations Dilations The first three are congruency (Isometry) transformations. In other words the new figure is congruent to the old figure (Pre-Image).

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Chapter 9

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  1. Chapter 9 Transformations

  2. 9.1 Reflections

  3. Types • There are four types of transformations: • Reflections • Translations • Rotations • Dilations • The first three are congruency (Isometry) transformations. In other words the new figure is congruent to the old figure (Pre-Image). • Dilations are similarity transformations. The new image is different in size from the pre-image.

  4. Reflections • Key words are “mirror image” or “flip” • You can reflect across a line or a point. • For Coordinate Geometry • Most common lines are the y-axis, x-axis, y = x line or any horizontal or vertical line. • Most common point that the pre-image is reflected across is the origin. • We will use matrices for coordinate geometry.

  5. Reflections across a line Line of Reflection Notice the new image is a “flip” of the pre-image across the line of reflection. A A’ B B’ If we pick two corresponding points on the pre-image and the reflected image, notice the line of reflection is the perpendicular bisector of the segments.

  6. Reflections across a line Line of Reflection What happens when the line of reflection goes through the pre-image? A’ A Notice the new image is a “flip” of the pre-image across the line of reflection. B B’ C C’ Notice the pre-image C is on the line of reflection, thus the new image C’ is in the exact place.

  7. Reflections across a point Notice the new image is a “flip” of the pre-image across the point of reflection. It looks very similar to a “rotation.” In 9.3 you will see why. Point of Reflection The point of reflection is the midpoint between any point on the pre-image and the new image.

  8. Coordinate Geometry • There are specific rules you need to memorize in order to do reflections in coordinate geometry. • Reflect across x axis – P(x,y)  P’(x, -y) • Reflect across y axis – P(x,y)  P’(-x, y) • Reflect across origin – P(x,y)  P’(-x, -y) • Reflect across y=x line P(x,y)  P’(y,x) • Notice which points become negative!

  9. Example • TakeΔABC where A(-3, 4), B(0, 8) and C(5, -2) • Ref across x axis: • A’(-3, -4), B’(0, -8) and C’(5, 2) • y’s change sign. • Ref across y axis: • A’(3, 4), B’(0, 8) and C’(-5, -2) • x’s change sign.

  10. Example Continued • TakeΔABC where A(-3, 4), B(0, 8) and C(5, -2) • Ref across origin: • A(3, -4), B(0, -8) and C(-5, 2) • Everything changes sign. • Ref across y = x line: • A(4, -3), B(8, 0) and C(-2, 5) • x’s and y’s change position.

  11. Review of Matrices • Matrices can be added, subtracted, scalar multiplied and multiplied. • You did the first three in algebra I. You must know the last one for this section. • The good part is that the TI – 83 does it for you without you having to know how manually. • The size of the matrix is the number of rows by the number of columns.

  12. Addition of Matrices • Matrices can only be added or subtracted if they are the same size. That is the same number of row and the same number of columns. 2x3 2x3 2x3

  13. Scalar Multiplication • Scalar Multiplication is very similar to distribution. You have a constant outside of the matrix multiplying the matrix by it.

  14. Multiplication • Multiplication is very intricate. All you will need to know how to do is plug the matrices into the calculator and multiply. • To multiply matrices the number of columns of the first matrix must equal the number of rows in the second matrix. • You can multiply a 2x3 by a 3x5 because the number of columns in the first (3) is equal to the number or rows in the second (3). • You can’t do the reverse.

  15. Multiplication • Matrix [A]= • Matrix [B]= • Find [A][B] =

  16. Multiplication (H) (2x3) by (3x1) = (2x1) (2)(2)+(3)(-3)+(-4)(1) = -9 (-1)(2)+(2)(-3)+(5)(1) = -3

  17. So why Matrices • If you’re giving coordinates for any polygon you can put those coordinates in matrix form. • For example ΔABC where A(-3, 4), B(0, 8) and C(5, -2) can be written in a 2x3 size matrix like this: Point A Point B Point C

  18. Reflections with Matrixes • Ref across x axis,multiply by this: • Ref across y axis,multiply by this: • Ref across origin,multiply by this: • Ref across y = x line, • Multiply by this:

  19. Reflections Across x axis • All you need to do is multiply the matrix that you will use for the reflection and the matrix that is for the polygon. • The result will be the new matrix.

  20. Reflections with Matrices • You must remember the order is important! • The first matrix is the reflection matrix • The second matrix is the matrix for the polygon. • If you mix the order of the matrices up, you will not be able to multiply them. • [A][B] is not always equal to [B][A]….

  21. 9.2 Translations

  22. Translations • The key word for translations is “slide” • You can translate “slide” a figure along a line. • The key point is the all corresponding points move the exact same distance. • This is also a congruency (Isometry) transformation.

  23. Translations Notice all segments are congruent to each other. It is the distance from corresponding points that are all the same. A’ C’ A B’ C B

  24. Compositions • Compositions are multiple transformations. • A Translation can be made by double reflections across parallel lines.

  25. Coordinate Geometry • You can translate (slide) either parallel to the x axis, parallel to the y axis or a do successive translations where you move along the x axis first, then the y axis. • Example of RULE: • To move a point 5 to the right and 2 down. P(x,y)  P’(x + 5, y – 2) • P(3, 5)  • P’( 3 + 5, 5 – 2)  • P’ (8, 3)

  26. Matrix Translations • Matrix Translations are the easiest of all the matrix transformations. • There is only one matrix to memorize and there is only addition. • Take Quadrilateral ABCD where A(-2, 5), B(0, 6), C(3, 0) and D(7, -1) and we want to move it 5 units to the right and 2 units down. • Remember the Rule: P(x,y)  P’(x+5, y-2)?

  27. Matrix Translations • Quadrilateral ABCD where A(-2, 5), B(0, 6), C(3, 0) and D(7, -1) • Original Matrix (Pre-Image) • Translation Matrix to move a quadrilateral “5 units to the right and 2 units down” • Translated Quadrilateral A’B’C’D’

  28. 9.3 Rotations

  29. Rotations • The third transformation is a Rotation. • The key word is “spin” • You will rotate (Spin) an object about a point. • A rotation is also another Isometry transformation. • This point is called the “center of rotation” • In coordinate geometry, it usually is the origin. • You will rotate the figure a certain number of degrees called the “angle of rotation”.

  30. Rotation Center of Rotation Rotations can be clockwise or counter clockwise.

  31. Rotations • Rotations can also be thought of as a composition transformation. • It is a double reflection across non-parallel lines. • The angle made between the non-parallel lines is ½ the angle of rotation.

  32. Double Reflection The 50° angle made between the intersection lines of reflection creates a 100° angle of rotation. 50°

  33. Coordinate Geometry • Coordinate Geometry rotations are performed with the origin as the center of rotation. • There are three matrices you need to memorize to do this. • Remember, when you multiply you must put one of these three matrices first then the matrix for the polygon second.

  34. Coordinate Geometry Rotate 90° CCW or 270°CW. Rotate 180° CCW or 180°CW. This is the same matrix as reflecting across a point. Rotate 270° CCW or 90°CW.

  35. Example Rotate Quadrilateral ABCD 90 degrees clockwise, where A(-2, 5), B(0, 6), C(3, 4) and D(7, -1)

  36. Example

  37. Example 90° CW rotation around origin.

  38. 9.5 Dilations

  39. Dilations • Dilations are the only transformations that are not Isometry. • They are similarity transformations. • So, the pre image and the new image are not the same size or location. • Dilations can be enlargements or reduction depending on the |r|. If |r|>1 you have an enlargement. If 0 < |r| < 1 you have a reduction. • You will have a center of dilation.

  40. Example r = 2 Since r = 2, the new location of the tail will be twice as far away from the center of dilation as the pre-image. Same thing for the new location of the eye. It will be twice as far away from the center of dilation as the pre-image. Center of Dilation What you end up with is a figure that is twice as large (b/c r = 2) as the pre-image AND twice as far away from the Center of Dilation.

  41. r = 3 What do you think will happen when r = 3? You have a figure that is 3 times as large and 3 times as far away from the Center as the pre-image. Center of Dilation

  42. Example r = 1/2 Since r = 1/2, the new location of the tail will be half as far away from the center of dilation as the pre-image. Same thing for the new location of the eye. It will be half as far away from the center of dilation as the pre-image. Center of Dilation What you end up with is a figure that is half as large (b/c r = 1/2) as the pre-image AND half as far away from the Center of Dilation.

  43. Dilations and Coordinate Geometry • Dilations are pretty easy with coordinate geometry and matrices. • You will need to do scalar multiplication of the matrix. • The scalar multiplier is your r!

  44. r = 2 • Dilate ΔABC where A(-3, 4), B(3, 3) and C(5, -2) with the origin as the center point. • Since r = 2, you will need to multiply the matrix by 2. 2

  45. Example

  46. Example

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