5-1: Transformations

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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
• 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>
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.