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)

Counterclockwise rotations are the norm 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°

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.

Turn in the following problems