Question

Part A: Write functions for each of the following transformations using function notation. Choose a different letter to represent each function. 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
Part B: How do these functions differ from functions you have used in algebra in the past?

Part C: Write a function to represent each series of transformations:

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

Answer 1

Transformation functions for translations, reflections, and rotations have been described. These operations alter the position of points in a coordinate plane, which is different from solving algebraic functions. Combined transformation functions have also been provided for specific sequences of transformations.

Step-by-step explanation:

Part A: Transformation Functions

Let's denote each transformation with a different letter as a function of the original coordinates (x, y). Here are the functions for each listed transformation:

• Translation by a units to the right and b units up: T(x, y) = (x + a, y + b)
• Reflection across the y-axis: Ry(x, y) = (-x, y)
• Reflection across the x-axis: Rx(x, y) = (x, -y)
• Rotation of 90 degrees counterclockwise: R90(x, y) = (-y, x)
• Rotation of 180 degrees counterclockwise: R180(x, y) = (-x, -y)
• Rotation of 270 degrees counterclockwise: R270(x, y) = (y, -x)

Part B: Differences from Algebraic Functions

These transformation functions differ from typical algebraic functions as they deal with geometric transformations rather than equations solving for one variable in terms of another. These functions represent operations on points in the coordinate plane, changing their positions through movements like translations, rotations, and reflections.

Part C: Combined Transformations

Here are the functions for each series of transformations:

• Rotation of 90 degrees counterclockwise about the origin, then a reflection across the x-axis: C1(x, y) = Rx∘R90(x, y) = (y, -x)
• Reflection across the y-axis, then a translation by a units to the right and b units up: C2(x, y) = T∘Ry(x, y) = (-(x + a), y + b)
• Translation by a units to the right and b units up, followed by rotation of 180 degrees counterclockwise about the origin, then a reflection across the y-axis: C3(x, y) = Ry∘R180∘T(x, y) = -((x + a), -(y + b)) = (-x - a, y + b)