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.

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

please use this format:
T(x, y) = (x + 6, y)
T<6, 0>(x, y) = (x + 6, y)
T6, 0(x, y) = (x + 6, y)

Answer 1

`(a)\ T_(a,b) =(x +a,y+b)`

`(b)\ F_(y\ axis) = (-x,y)`

`(c)\ F_(x\ axis) = (x,-y)`

`(d)\ R_(o,90^o) = (-y,x)`

`(e)\ R_(o,180^o) = (-x,-y)`

`(f)\ R_(o,270^o) = (y,-x)`

Explanation:

Solving (a): Translate a units right, b units up

When a function is translated a units right, the number of units will be added to the x coordinate

When a function is translated b units up, the number of units will be added to the y coordinate.

So, we have:

`T_(a,b) =(x +a,y+b)`

Solving (b): Reflect across y-axis

When a function is translated across the y-axis, the x coordinate gets negated.

So, we have:

`F_(y\ axis) = (-x,y)`

Solving (c): Reflect across x-axis

When a function is translated across the x-axis, the y coordinate gets negated.

So, we have:

`F_(x\ axis) = (x,-y)`

Solving (d): 90 degrees rotation counterclockwise

When a function is rotated 90 degrees counterclockwise, the y-coordinates gets negated and then swapped with the x-coordinate.

So, we have:

`R_(o,90^o) = (-y,x)`

Solving (e): 180 degrees rotation counterclockwise

When a function is rotated 180 degrees counterclockwise, the coordinates are negated.

So, we have:

`R_(o,180^o) = (-x,-y)`

Solving (f): 270 degrees rotation counterclockwise

When a function is rotated 270 degrees counterclockwise, the x-coordinates gets negated and then swapped with the y-coordinate.

So, we have:

`R_(o,270^o) = (y,-x)`