How to Determine the Reflection of a Parabola by a Given Line?

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Discussion Overview

The discussion revolves around determining the algebraic reflection of a parabola described by the equation $$y^2-2y-4x-11=0$$ across the line $$y = -x$$. Participants explore both graphical and algebraic methods for this reflection.

Discussion Character

  • Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant requests an algebraic method for reflecting the parabola across the line $$y = -x$$, indicating familiarity with the graphical approach.
  • Another participant suggests that the reflection of points (1,0) and (0,1) results in the points (0,-1) and (-1,0), proposing that the image of a point (x,y) is given by (-y,-x).
  • A subsequent post questions the algebraic transformation $$(-x)^2-2(-x)-4(-y)-11=0$$, seeking clarification on its validity.
  • Another participant confirms the correctness of the transformation presented in the previous post.
  • A final post presents a series of equations, including the original parabola, the transformed equation, and the line of reflection, but does not elaborate further.

Areas of Agreement / Disagreement

There is no clear consensus on the algebraic method for reflection, as participants express different aspects of the problem without resolving the overall approach.

Contextual Notes

The discussion does not clarify the assumptions or definitions used in the transformation process, and the mathematical steps remain unresolved.

Monoxdifly
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Determine the reflection of a parabola $$y^2-2y-4x-11=0$$ by the line y = -x.

I know how to do it graphically, but please tell me how to do it algebraically.
 
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Monoxdifly said:
Determine the reflection of a parabola $$y^2-2y-4x-11=0$$ by the line y = -x.

I know how to do it graphically, but please tell me how to do it algebraically.
In such a reflection, the images of the points (1,0) and (0,1) are (0, -1) and (-1,0), respectively.

This means that the image of (x,y) is (-y, -x). You only need to substitute that in the equation.
 
$$(-x)^2-2(-x)-4(-y)-11=0$$?
 
Monoxdifly said:
$$(-x)^2-2(-x)-4(-y)-11=0$$?
Yes
 
[DESMOS=-20,20,-13.35559265442404,13.35559265442404]y^2-2y-4x-11=0;x^2+2x+4y-11=0;y=-x;[/DESMOS]
 

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