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

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:

https://brainly.com/question/12865301

ACCESS MORE
EDU ACCESS