Respuesta :
In geometry, transformations are used to move points from one position to another. The functions to represent the transformations are as follows:
- [tex]S(90,0)/R_x(x,y) = (-y,-x)[/tex] .
- [tex]R_y/<a,b>(x,y) = (-x-a,y+b)[/tex].
- [tex]<a,b>(x,y)/S(180,0)/R_y = (x+a,-y-a)[/tex]
(a) Rotation of 90 degrees counterclockwise about the origin, point o, then a reflection across the x-axis
When a point is rotated [tex]90^o[/tex] counterclockwise, the rule is:
[tex](x,y) \to (-y,x)[/tex]
The rule of reflection across the x-axis is:
[tex](x,y) \to (x,-y)[/tex]
So, the combined transformation is:
[tex](-y,x) \to (-y,-x)[/tex]
Hence, the function that represents the transformation is:
[tex]S(90,0)/R_x(x,y) = (-y,-x)[/tex]
(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:
[tex](x,y) \to (-x,y)[/tex]
The rule of translation, a units right and b units up is:
[tex](x,y) \to (x + a,y+b)[/tex]
So, the combined transformation is:
[tex](-x,y) \to (-x - a,y+b)[/tex]
Hence, the function that represents the transformation is:
[tex]R_y/<a,b>(x,y) = (-x-a,y+b)[/tex]
(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:
[tex](x,y) \to (x + a,y+b)[/tex]
When a point is rotated [tex]180^o[/tex] counterclockwise, the rule is:
[tex](x,y) \to (-x,-y)[/tex]
The combined transformation at this point is:
[tex](x + a, y + a) \to (-x-a,-y-a)[/tex]
The rule of reflection across the y-axis is:
[tex](x,y) \to (-x,y)[/tex]
So, the overall transformation is:
[tex](-x-a,-y-a) \to (x+a,-y-a)[/tex]
Hence, the function that represents the transformation is:
[tex]<a,b>(x,y)/S(180,0)/R_y = (x+a,-y-a)[/tex]
Read more about transformation of points at:
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