Question

Functions and Transformations:

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
A) T(x, y) = (x + a, y + b)
B) R(x, y) = (x - a, y - b)
C) S(x, y) = (x - a, y + b)
D) N/A

Reflection across the y-axis
A) Y(x, y) = (-x, y)
B) X(x, y) = (x, -y)
C) Z(x, y) = (-x, -y)
D) N/A

Reflection across the x-axis
A) X(x, y) = (x, -y)
B) Y(x, y) = (-x, y)
C) Z(x, y) = (-x, -y)
D) N/A

Rotation of 90 degrees counterclockwise about the origin, point O
A) O(x, y) = (y, -x)
B) P(x, y) = (-y, x)
C) Q(x, y) = (-x, -y)
D) N/A

Rotation of 180 degrees counterclockwise about the origin, point O
A) O(x, y) = (-x, -y)
B) P(x, y) = (-y, -x)
C) Q(x, y) = (-x, -y)
D) N/A

Rotation of 270 degrees counterclockwise about the origin, point O
A) O(x, y) = (-y, x)
B) P(x, y) = (y, -x)
C) Q(x, y) = (-x, -y)
D) N/A

Part B: How do these functions differ from functions you have used in algebra in the past?
A) They involve geometric transformations.
B) They use different variables.
C) They represent movements in a coordinate plane.
D) N/A

Part C: In the same way that other functions can be combined, a series of transformations can be combined into a single function.

Rotation of 90 degrees counterclockwise about the origin, point O, then a reflection across the x-axis
A) R(x, y) = (-y, -x)
B) S(x, y) = (-x, -y)
C) T(x, y) = (-y, x)
D) N/A

Reflection across the y-axis, then a translation a units to the right and b units up
A) U(x, y) = (a - x, b - y)
B) V(x, y) = (x - a, y - b)
C) W(x, y) = (x + a, y + b)
D) N/A

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
A) X(x, y) = (-x - a, -y - b)
B) Y(x, y) = (-x + a, -y + b)
C) Z(x, y) = (-y + a, x + b)
D) N/A

Answer 1

Part A: The functions for each transformation are provided. Part B: The functions differ as they involve geometric transformations and use different variables. Part C: The functions for combined transformations are given.

Step-by-step explanation:

Part A:

• A) The translation of a units to the right and b units up is represented by the function T(x, y) = (x + a, y + b).
• B) There is no function provided for this transformation.
• C) The reflection across the y-axis is represented by the function Y(x, y) = (-x, y).
• D) There is no function provided for this transformation.
• A) The reflection across the x-axis is represented by the function X(x, y) = (x, -y).
• B) There is no function provided for this transformation.
• A) The rotation of 90 degrees counterclockwise about the origin is represented by the function O(x, y) = (y, -x).
• B) There is no function provided for this transformation.
• A) The rotation of 180 degrees counterclockwise about the origin is represented by the function O(x, y) = (-x, -y).
• B) There is no function provided for this transformation.
• A) The rotation of 270 degrees counterclockwise about the origin is represented by the function O(x, y) = (-y, x).
• B) There is no function provided for this transformation.

Part B:

The functions in this question differ from functions used in algebra in the past because they involve geometric transformations and represent movements in a coordinate plane using different variables.

Part C:

The combination of a rotation of 90 degrees counterclockwise about the origin, then a reflection across the x-axis is represented by the function R(x, y) = (-y, -x). The combination of a reflection across the y-axis, then a translation a units to the right and b units up is represented by the function V(x, y) = (x - a, y - b). The combination of a 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 is represented by the function X(x, y) = (-x - a, -y - b).