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Unit 8: Figures and Transformations

Section 1: Graphing and Naming Figures In Unit 4 section 1 we discussed graphing figures and translating those graphs Points are named with capital letters Image points are named with a capital letter and prime notation (apostrophe) Polygons to know:

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Unit 8: Figures and Transformations

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  1. Section 1: Graphing and Naming Figures • In Unit 4 section 1 we discussed graphing figures and translating those graphs • Points are named with capital letters • Image points are named with a capital letter and prime notation (apostrophe) • Polygons to know: • triangle = 3 sides quadrilateral = 4 sides • pentagon = 5 sides hexagon = 6 sides • heptagon = 7 sides octagon = 8 sides • nonagon = 9 sides decagon = 10 sides Unit 8: Figures and Transformations

  2. When naming a polygon, list the letters in order as you move around the figures (usually in alphabetical order) • For triangles, draw a small triangle before the letters • Some types of quadrilaterals to know: • Square: 4 equal sides and 4 equal angles • Rectangle: 2 pairs of equal sides and 4 equal angles • Trapezoid: 1 pair of parallel sides • Parallelogram: 2 pair of equal sides and opposite angles are equal

  3. Rhombus: 4 equal sides and opposite angles are equal • To find the total number of degrees in a polygon: (n – 2) · 180° where n is the number of sides • Area of a triangle = ½bh where b = base and h = height (base and height must meet at a 90° angle) • Area of a square = s² where s = length of side • Area of a rectangle = lw where l = length and w = width • Area of a trapezoid = where h = height and b1 and b2 are the bases

  4. Ex1. A = (-2, 5), B = (3, 6) and C = (1, -2) a) Graph b) Graph T-3,-2 • Ex2. G = (-3, 4), H = (2, 4), I = (2, -3) and J = (-3, -3) a) graph b) name the type of figure c) find the area

  5. Section 2: Size Changes • Another type of transformation is a size change • Use the prime notation for the image • With a size change, you will be multiplying each coordinate of the point by the size change factor • If the absolute value of the size change factor is larger than 1, the graph will get larger and it is called an expansion • If the absolute value of the size change factor is between 0 and 1, the graph will get smaller and is called a contraction • The size change factor a.k.a. the magnitude of the size change

  6. You can write size changes in a few ways: S(x, y) = (kx, ky) or Sk or size change of magnitude k • Ex1. A = (-2, 4), B = (3, 1), C = (5, -2), D = (-1, -3) a) graph b) graph S-2 • The negative number inverts the figure

  7. The image and the preimage under a size change are similar figures • Ex2. X = (-8, 6), Y = (2, -2), Z = (-4, -4). Find the coordinates of the image if the figure is increased by 40%. Is the image a contraction or expansion? • When a person earns time and a half, it means they make 1.5 times their normal hourly wage. • Ex3. When working overtime, Karen earns time and a half. At that time she earns $21.75 an hour. What is her normal wage? • Section of the book to read : 6-7

  8. Section 3: Scale Changes • Scale changes are like size changes, but the x and y-values are multiplied by different numbers • Ways to write scale changes: S(x,y) = (ax, by) or Sa,b • The preimage and image are NOT similar figures under a scale change • Negative numbers in the scale change “flip” the figure either vertically or horizontally, depending in which number is negative

  9. Ex1. A = (-2, 3), B = (2, 5), C = (4, -2), D = (-4, -4) a) graph ABCD b) graph ABCD under S(x, y) = (½x, 2y)

  10. Section 4: Reflection over the x-axis • To reflect over a line means to “flip” the figure over the line (in this case the x-axis) • When you are reflecting a figure, reflect each individual point over the line • Rename the point using prime notation • Reflections are written with a lower case r and then the axis (or line) it is being reflected over as the subscript rx • The image and preimage are congruent figures

  11. Ex1. Reflect each individual point over the x-axis and name the image • A) (-2, 5) • B) (3, 2) • C) (-3, -4) • Ex2. What do you notice happened to each point?

  12. Ex3. J = (-2, -5), K = (3, -2), L = (4, 1), M = (-3, 4) a) Graph JKLM b) Graph JKLM under rx • Ex4. Without graphing, give the coordinates of A = (-2, -4), B = (3, -2), C = (1, 5) after rx

  13. Section 5: Reflection over the y-axis • Reflect each point over the y-axis and rename using prime notation • We denote a reflection over the y-axis as ry • The image and preimage are congruent figures • Just like with the reflection over the x-axis, you will notice a simple pattern with one of the coordinates that will allow you to reflect quickly and without graphing

  14. Ex1. A = (3, -2), B = (4, 3), C = (-3, 1), D = (-2, -4) a) Graph ABCD b) Graph ABCD under ry • Ex2. Without graphing, find the coordinates of the image points of X = (-2, 4), Y = (-6, 9), Z = (3, 5) under ry

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