Question

Transformations can be thought of as functions that take the points from one object (the pre-image) to the corresponding points on another object (the image). In the unit, you saw examples of how algebra can be used to rewrite the coordinates of points after they have been transformed. You can also use function notation to rewrite the coordinates. For example, these statements show several different ways you can use function notation for the translation, T, of a point 6 units in the positive x-direction:
T(x, y) = (x + 6, y)
T:(x, y) = (x + 6, y)
T<6, 0>(x, y) = (x + 6, y)
T6, 0(x, y) = (x + 6, y)
Part A
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.

translation of a units to the right and b units up
reflection across the y-axis
reflection across the x-axis
rotation of 90 degrees counterclockwise about the origin, point O
rotation of 180 degrees counterclockwise about the origin, point O
rotation of 270 degrees counterclockwise about the origin, point O

Answer 1

1. Translation of a units to the right and b units up has rule:

(x,y)→(x+a,y+b).

Then the function is

$T(x,y)=(x+a,y+b).$

2. Reflection across the y-axis acts in following way:

(x,y)→(-x,y).

Then the function is

$R_y(x,y)=(-x,y).$

3. Reflection across the x-axis acts in following way:

(x,y)→(x,-y).

Then the function is

$R_x(x,y)=(x,-y).$

4. Rotation of 90 degrees counterclockwise about the origin, point O acts bu the rule:

(x,y)→(-y,x).

Then the function is

$R_{O,90^{\circ}}(x,y)=(-y,x).$

5. Rotation of 180 degrees counterclockwise about the origin, point O acts bu the rule:

(x,y)→(-x,-y).

Then the function is

$R_{O,180^{\circ}}(x,y)=(-x,-y).$

6. Rotation of 270 degrees counterclockwise about the origin, point O acts bu the rule:

(x,y)→(y,-x).

Then the function is

$R_{O,270^{\circ}}(x,y)=(y,-x).$