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Transformations – Translations and Reflections

Transformations – Translations and Reflections. HW #1. Defining a Transformation. A transformation is an operation that maps , or moves, a figure onto an image. What is a RIGID or ISOMETRIC transformation?.

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Transformations – Translations and Reflections

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  1. Transformations –Translations and Reflections HW #1

  2. Defining a Transformation A transformation is an operation that maps, or moves, a figure onto an image.

  3. What is a RIGID or ISOMETRIC transformation? A rigid/isometric transformation does not alter the size or shape of the original object. How can you move an object without changing its size or shape?

  4. Defining a Translation A translation is a “slide” of an object on the coordinate plane. It can be described using words: Up, Down, Left, Right

  5. Defining a Translation A translation can be described as a function using coordinates: ( x , y )  ( x + a , y + b) The amount of up/down movement The amount of left/right movement

  6. Defining a Translation Using this notation, which part describes the input and the output? ( x , y )  ( x + a , y + b) Example: Given the point ( -9 , 2 ), where would the image point be given the following translation? ( x , y )  ( x - 5 , y +8)

  7. Example 2 Describe the translation of the segment. SOLUTION Point P is moved 4 units to the right and 2 units down to get to point P'. So, every point on PQ moves 4 units to the right and 2 units down. Describe Translations *Notice the notation to differentiate between the original points, and the image points!*

  8. Example 3 The translation can be described using the notation (x, y)(x – 3, y + 4). ANSWER Use Coordinate Notation Describe the translation using coordinate notation. Did the triangle change shape or size?

  9. Defining a Reflection A reflection is an isometrictransformation that creates a mirror image. The original figure is reflected over a line that is called the line of reflection.

  10. Checkpoint Identify Reflections Tell whether the red figure is a reflection of the blue figure. If the red figure is a reflection, name the line of reflection. 1. 3. 2. ANSWER ANSWER ANSWER yes; the x-axis yes; the y-axis no

  11. Line of Reflectional Symmetry If a figure can be reflected onto itself, then it has a line of reflectional symmetry.

  12. Example 4 Determine Lines of Symmetry Determine the number of lines of symmetry in a square. SOLUTION Think about how many different ways you can fold a square so that the edges of the figure match up perfectly. vertical fold horizontal fold diagonal fold diagonal fold A square has four lines of symmetry. ANSWER

  13. Checkpoint Determine Lines of Symmetry Determine the number of lines of symmetry in the figure. 1. 1 ANSWER 2. 2 ANSWER 3. 4 ANSWER

  14. Multiple translations Sketch ABC given A(1, 3), B(4, 3), and C(0, 6). First, reflect the triangle over the x-axis. Label this image A’B’C’. Next, take the new image and translate it such that (x, y) (x – 4, y + 3). Label this new image as A’’B’’C’’.

  15. Transformations – Rotations HW #2

  16. Defining a Rotation A rotation is a transformation in which a figure is turned about a fixed point. The fixed point is the center of rotation. The angle of rotation is the amount that the object has been rotated.

  17. Defining a Rotation Does rotating a figure change its size or shape? A rotation is an ISOMETRIC transformation.

  18. Defining a Rotation How many degrees are in one full rotation? How many degrees are in a “half turn”? How many degrees are in a “quarter turn”? Directions of rotations clockwise counter-clockwise

  19. More degrees of rotation To find one angle of rotation, divide 360 by the number of sides: Ex: Hexagon, 6 sides

  20. Sketching a Rotation Rotate the quadrilateral wxyza half turn around point y.

  21. Sketching a Rotation Rotate the following quadrilateral a quarter turn clockwise around point c. a b d e c f

  22. Checkpoint Ex. Sketch the triangle with vertices A(0, 0),B(3, 0),andC(3, 4).Rotate∆ABC 90°counterclockwise about the origin. Name the coordinates of the new vertices A',B', and C'. ANSWER A'(0,0), B'(0, 3), C'(–4, 3) Rotations in a Coordinate Plane

  23. Defining Rotational Symmetry A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180o or less.

  24. a. Rectangle b. Regular hexagon c. Trapezoid Example 1 Identify Rotational Symmetry Does the figure have rotational symmetry? If so, describe the rotations that map the figure onto itself. Number of degrees the figure can rotate:

  25. Checkpoint 3. Regular octagon yes; a clockwise or counterclockwise rotation of 45°, 90°, 135°, or 180° about its center ANSWER Identify Rotational Symmetry

  26. 1.1/1.2 – Patterns and Inductive Reasoning Objectives #4 and 6 HW: #3 *both parts Mini Quiz Wednesday

  27. Why Learn This? • In your home life and through your job, you will look for patterns in an attempt to draw conclusions or make predictions. • A conjecture is an unproven statement that is based on a pattern or observation. • “I think the longer we wait, the cheaper a new iPhone will be.” • Inductive Reasoningis the process of looking for patterns and making conjectures.

  28. Find the next term in the pattern: Examples: 1. 2. 3.

  29. Forming conjectures using inductive reasoning: Examples: Find the next two terms in the sequence and determine the pattern. 5, 8, 12, 17, 23, … 2, 6, 18, 54, 162, …

  30. Example 1 1 + 1 = 2 5 + 1 = 6 3 + 7 = 10 3 + 13 = 16 21 + 9 = 30 101 + 235 = 336 The sum of any two odd numbers is even. ANSWER Make a Conjecture Complete the conjecture. ? Conjecture:The sum of any two odd numbers is ____. Begin by writing several examples. Each sum is even. You can make the following conjecture.

  31. Conjecture Example: Make a conjecture about the sum of the first 8 odd numbers. 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 The sum of the first 8 odd numbers will form a perfect square.

  32. Use inductive reasoning to make conjectures about the following sequences: 384, 192, 96, 48, … 3, 12, 48, 192, … M, T, W, T, …

  33. More than one reasonable answer? Find the next two terms in the pattern: 1, 1, 2, … There can be more than one logical way to perceive a pattern!

  34. Proving a Conjecture False • Just because something is true for several examples does not prove that it is true in general. • Ex) All fruits are red  Strawberries, Raspberries, Red Delicious Apples, Red Grapes • A conjecture is considered false if it is not always true. • To prove a conjecture false, you need one counterexample.

  35. Counterexample Examples: Any three points can be connected to form a triangle. Counterexample: The square of any number is greater than the original number. Counterexample:

  36. Counterexample Examples: The sum of two numbers is always greater than the larger of the two numbers. Counterexample: All shapes with four sides the same length are squares Counterexample:

  37. Find a counterexample for the following: • If an number is divisible by 5, then it is divisible by 10. • Subtracting a number will always result in a smaller quantity than the original value. • I’ve noticed that all of my healthy friends eat oranges. Therefore, if I eat oranges I’ll be healthy.

  38. Closure • What is inductive reasoning based on? • What is a counterexample? • Given the following sequence, write 2 reasonable ways to continue the pattern. • 1, 2, 1, ___, ____,

  39. 1.3 Points, Lines, and Planes Homework: Assignment #5

  40. Segment: part of a line with two endpoints Ray: Part of a line with only one endpoint Opposite Rays: Two collinear rays with the same endpoint A B C D R E S

  41. Collinear: Noncollinear: Coplanar: Noncoplanar: Opposite Rays: Intersection:

  42. RAYS ARE DIRECTIONAL!! R Q QR R Q RQ R Q

  43. Lines that do not intersect: Parallel: coplanar lines that do not intersect

  44. Skew: noncoplanar lines that are not parallel and do not intersect

  45. Name AB in three other ways. Name a point that is collinear with F. Name a point that is on l and m.

  46. Name the plane at the bottom of the cube. Name a line in plane GEF. Name a point coplanar with E and F. Name a line coplanar with EA.

  47. Postulate/Axiom – an accepted statement of fact. Through any two points there is exactly one line. Through any three noncollinear points there is exactly one plane. A C B

  48. If two lines intersect, then they intersect in exactly one point. If two planes intersect, then they intersect in exactly one line.

  49. If two points lie in a plane, then the line containing those points lies in the plane. W A B

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