Question

Part C
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, 0)/<-3,1>(x, y) = (y - 3, -x + 1).
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

In geometry, transformations are used to move points from one position to another. The functions to represent the transformations are as follows:

• $S(90,0)/R_x(x,y) = (-y,-x)$ .
• $R_y/<a,b>(x,y) = (-x-a,y+b)$.
• $<a,b>(x,y)/S(180,0)/R_y = (x+a,-y-a)$

(a) Rotation of 90 degrees counterclockwise about the origin, point o, then a reflection across the x-axis

When a point is rotated $90^o$ counterclockwise, the rule is:

$(x,y) \to (-y,x)$

The rule of reflection across the x-axis is:

$(x,y) \to (x,-y)$

So, the combined transformation is:

$(-y,x) \to (-y,-x)$

Hence, the function that represents the transformation is:

$S(90,0)/R_x(x,y) = (-y,-x)$

(b) Reflection across the y-axis, then a translation a units to the right and b units up

The rule of reflection across the y-axis is:

$(x,y) \to (-x,y)$

The rule of translation, a units right and b units up is:

$(x,y) \to (x + a,y+b)$

So, the combined transformation is:

$(-x,y) \to (-x - a,y+b)$

Hence, the function that represents the transformation is:

$R_y/<a,b>(x,y) = (-x-a,y+b)$

(c) 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

The rule of translation, a units right and b units up is:

$(x,y) \to (x + a,y+b)$

When a point is rotated $180^o$ counterclockwise, the rule is:

$(x,y) \to (-x,-y)$

The combined transformation at this point is:

$(x + a, y + a) \to (-x-a,-y-a)$

The rule of reflection across the y-axis is:

$(x,y) \to (-x,y)$

So, the overall transformation is:

$(-x-a,-y-a) \to (x+a,-y-a)$

Hence, the function that represents the transformation is:

$<a,b>(x,y)/S(180,0)/R_y = (x+a,-y-a)$

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