Geometry Homework

Question

Write functions for each of the following transformations using function notation. Choose a different letter to represent each function. For example, you can use R to represent rotations. Assume that a positive rotation occurs in the counterclockwise direction.

1. translation of a units to the right and b units up
2. reflection across the y-axis
3. reflection across the x-axis
4. rotation of 90 degrees counterclockwise about the origin, point O
5. rotation of 180 degrees counterclockwise about the origin, point O
6. rotation of 270 degrees counterclockwise about the origin, point O
7. T(x, y) = (x + 6, y)
8. T:(x, y) = (x + 6, y)
9. T<6, 0>(x, y) = (x + 6, y)
10. T6, 0(x, y) = (x + 6, y)

Answer 1

1. Letter representation:

There are actually standard letters used as symbols for transformations in math. Since your direction is to use a different letter for each, I will suggest using M for reflection since they make mirror images (normally it is a lower case r).

So...

R= rotation

M = reflection

T= translation

2.Transformation definitions:

These are RIGID transformations, which means the size and shape will NOT change, just the location of the point, line, or figure.

Translation is a slide left or right, up or down.

Reflection is a mirror image over a line of reflection.

Rotation is a turn forward or backwards about a certain point. When rotated in increments of 90°, each 90 degrees represent 1 turn, or a movement 1 quadrant forward or backwards.

3. Function Notation Rules:

General transformation notaion rules:

1. You must literally "note" the changes that are occurring AND represent that with an ordered pair in the form (x, y).

1. Your transformation letter tells you what transformation is happening.
2. The ordered pair tells you the actual rule or movement.
3. Positive and negative values of x and y in an ordered pair do NOT mean that they have positive or negative value. + and - tell you that x and y will switch signs when their transformation occurs.

Translation basics:

1. Remember that x always follows the x axis so represents left to right movement (horizontal) and y always follows the y axis so represents up and down movement (vertical).
2. Movement right and up are positive movements.
3. Movement left and down are negative movements.

Reflection basics:

1. Must indicate WHERE something is being reflected (X axis, y axis, across a line).
2. Must show sign for movement of x and y in the ordered pair

Rotation basics:

1. Must tell you the location where the rotation is happening (at the origin, about a different point)
2. Must tell you the number of degrees to rotate
3. Must tell you the direction to rotate (clockwise or counterclockwise)
4. Must show sign for the movement of x and y in the order pair.

4. Solutions:

Translation of a units to the right and b units up:

T_a,b → (x + a, y + b)

[This is read as "Translation of positive a along the x axis and poitive b along the y axis" or more simply, "Translation of a units to the right and b units up" as your question states. ]

Reflection across the y- axis

This is a left to right movement so the sign of your x value is changing but the y value stays the same.

M_y-axis (x, y) → (-x, y)

Reflection across the x-axis

This is an up and down movemet so so the sign of your y is changing but the x value stays the same.

M_x-axis (x, y) → (x, -y)

Rotation of 90 counterclockwise about the origin, point 0

This is "1" turn, so it moves 1 quadrant, which means your values "flip"

R_{o, 90°} (x, y) → (-y, x)

Rotation of 180 degrees counterclockwise across the origin, point 0

This is "2" turns,so it moves 2 quadrants which means x and y will have opposite signs after the move.

R_{o, 90°} (x,y)→ (-x, -y)

Rotation of 270 degrees counterclockwise across the origin, point 0

This is "3" turns, so it moves 3 quadrants which means your values will "flip"

R_{o, 270°} (x,y) → (-y, x)