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

Question

In the same way that other functions can be combined, a series of transformations can be combined into a single function. For example, this statement for the function S shows one way to represent the rotation of a point 270° counterclockwise about the origin followed by a translation 3 units to the left and 1 unit up: S(270, O)/<-3,1>(x, y) = (y – 3, -x + 1).

Write a function to represent each series of transformations:

1. rotation of 90 degrees counterclockwise about the origin, point O, then a reflection across the x-axis
2. reflection across the y-axis, then a translation a units to the right and b units up
3. 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

1) For a rotation about the origin of 90 degrees, the line segment connecting the transformed point with the origin will be perpedicular with the line segment connecting the initial point with the origin. For a point with coordinates (x,y), the slope of the line segment connecting this point with the origin will be m_i- = y/x according to the Rise Over Run rule, so the slope of the line segment connecting the origin and the transformed point will be m_t = -x/y, the negative reciprocal of the initial slope according to the Perpendicular Slope Rule.

For a counterclockwise rotation, a point initially in Quadrant I with coordinates (x, y) will end up in Quadrant II with coordinates (-y, x), retaining the distance from the origin an achieving that perpendicular slope.

A reflection across the x-axis will yield the same x-coordinates and the negative of the initial y-coordinates, transforming the point (-y, x) into (-y, -x)

Therefore, the transformation function takes the form f(x,y)=(-y, -x)

2) Reflection across the y-axis yields the same y-coordinates and the opposite of the x-coordinates, transforming (x, y) into (-x, y)

Translation a units to the right adds those units to the x-value of the ordered pair, while translation b units up adds those units to the y-value. The ordered pair (-x, y) takes the form (-x+a, y+b) after the specified translation.

Therefore, the transformation function takes the form f(x,y)=(-x+a, y+b)

3) Translation a units to the right and b units up, as explained in the previous part, transforms the ordered pair (x, y) into (x+a, y+b)

Rotation of 180 degrees, both clockwise and counterclockwise, yields the opposite value for both the x and y coordinates. This transforms (x+a, y+b) into (-x-a, -y-b).

Reflection across the y-axis yields the same y-coordinates and the opposite of the x-coordinates, transforming (-x-a, -y-b) into (x+a, -y-b)

Therefore, the transformation function takes the form f(x,y)=(x+a, -y-b)