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5-1: Transformations. English Casbarro Unit 5. Isometries. An isometry is a transformation that preserves both size and shape Also called a congruence transformation Reflections, translations and rotations are isometries Dilations are NOT isometries.
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5-1: Transformations English Casbarro Unit 5
Isometries • An isometry is a transformation that preserves both size and shape • Also called a congruence transformation • Reflections, translations and rotations are isometries • Dilations are NOT isometries
How to change points to show reflections (a flip of the figure) • Reflection across the y-axis: (a, b) (–a, b) • Reflection across the x-axis: (a, b) (a, –b) • Reflection across the line y = x: (a, b) (b, a) The line of symmetry is the line where a fold would match up both sides exactly. Ex. A figure with vertices (2,3), (–1, 4), and (0, 2) is reflected across the x axis, State the points of the new figure. Answer: (2, –3), (–1, –4), and (0, –2)
How to change the points to show translations • To show how a figure is translated on the coordinate plane, you will add or subtract the moves to the coordinate values: (a, b) (a + x, b + y) Ex. A figure with vertices (2,3), (–1, 4), and (0, 2) is translated 4 units to the right and 3 units down. Answer: You will add 4 to all of the x values, and subtract 3 from all of the y values. (2+4, 3-3), (–1+4, 4-3), and (0+4, 2-3) (6, 4), (3, 1), and (4, –1)
Notation to show translations • Ex. What is the translation of (3,4) under the translation (x, y) (x – 2, y + 7)? • Ex. What is the translation of (3,4) by the vector a = <-2, 7>
Reflecting across parallel lines will produce a translation.
How to change the points to show counterclockwise rotations • To show a 90° rotation: (a, b) (–b, a) • To show a 180° rotation: (a, b) (–a, –b) • To show a 270° rotation: (a, b) (b, –a) • To show a 360° rotation: (a, b) (a, b)
If it says clockwise rotation, change the measure into a counterclockwise rotation to use your rules.90° clockwise is the same as 270°counterclockwise, so you’d use the rules for 270° Counterclockwise rotations are the norm
A figure PQRST has the vertices (–1, –1), (–4, 1 ), (–2, 4), (0, 4), and (2, 1). • Find the new vertices under a rotation of 180° counterclockwise about • the origin. • 2. Find the new points under the translation (x, y)(x – 5, y + 2), then a • rotation 90° counterclockwise about the origin.
How to change the points to show dilations • To show all dilations and reductions: (a, b) (ka, kb) where k is the scale factor of the dilation. • Dilations require a center point and a scale factor.
Ex. A figure PQRST has the vertices (–1, –1), (–4, 1 ), (–2, 4), (0, 4), and (2, 1). • Find the vertices after a 180° rotation counterclockwise about the origin, • then a dilation by a scale factor of –2.
Standard Form of a Circle: Where the center is at (0,0), and r is the radius of the circle. EX 1: Here the circle has the center at (0,0) with a radius of 5 EX 2: Here the circle has the center at (4, –2) with a radius of 5. EX 3: Here the circle has the center at (–3 , –7) with a radius of 9. .
Solving Non-Linear Systems Example: Solve x2 + y2 = 25 x – y = –7
Solving Non-Linear Systems Example: Solve y = x2 + 3x + 2 y = 2x + 3 This is what the graph looks like. You can estimate the solution by the graph, but if You solve the problem, you can find the exact solution.
