5-1: Transformations 5-1: Transformations 5-1: Transformations 5-1: Transformations English CasbarroEnglish Casbarro

Unit 5Unit 5

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

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.

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)

How to change the points to show counterclockwise rotations

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). 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). 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:Standard Form of a Circle:

2 2( 4) ( 2) 25x y

2 2 25x y

2 2( 3) ( 7) 81x y

Where the center is at (0,0), and r Where the center is at (0,0), and r is the radius of the circle.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..

2 2 2x y r

Solving Non-Linear SystemsExample: Solve x2 + y2 = 25 x – y = –7

Solving Non-Linear SystemsExample: Solve y = x2 + 3x + 2 y = 2x + 3

This is what the graph looks like. You canestimate the solution by the graph, but ifYou solve the problem, you can find the exact solution.

Turn in the following problems