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

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In summary, the reflection of the parabola y^2-2y-4x-11=0 by the line y = -x can be determined algebraically by substituting (-y, -x) for (x,y) in the equation and solving for x and y.
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Monoxdifly
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Determine the reflection of a parabola \(\displaystyle y^2-2y-4x-11=0\) 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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  • #2
Monoxdifly said:
Determine the reflection of a parabola \(\displaystyle y^2-2y-4x-11=0\) 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.
 
  • #3
\(\displaystyle (-x)^2-2(-x)-4(-y)-11=0\)?
 
  • #4
Monoxdifly said:
\(\displaystyle (-x)^2-2(-x)-4(-y)-11=0\)?
Yes
 
  • #5
[DESMOS=-20,20,-13.35559265442404,13.35559265442404]y^2-2y-4x-11=0;x^2+2x+4y-11=0;y=-x;[/DESMOS]
 

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

What is a reflected parabola?

A reflected parabola is a type of parabola that is created when a parabola is reflected over a line. This line is called the axis of reflection and it is perpendicular to the axis of the parabola. The reflected parabola will have the same shape as the original parabola, but it will be flipped over the axis of reflection.

What is the equation for a reflected parabola?

The equation for a reflected parabola can be written in the standard form of y = a(x - h)^2 + k, where the values of a, h, and k determine the shape and position of the parabola. The axis of reflection is the line x = h and the vertex is located at the point (h, k).

How do you graph a reflected parabola?

To graph a reflected parabola, start by finding the vertex of the parabola. This can be done by finding the values of h and k in the equation y = a(x - h)^2 + k. Then, plot the vertex on the graph. Next, find two other points on the parabola by substituting different values for x into the equation. Plot these points on the graph and connect them to create the parabola. Finally, use the axis of reflection to flip the parabola over and complete the graph.

What are some real-life examples of reflected parabolas?

Reflected parabolas can be seen in many real-life situations. One common example is the shape of a satellite dish, which is a reflected parabola that helps to focus and reflect signals. Other examples include the shape of a parabolic mirror used in telescopes, the trajectory of a ball thrown off a cliff, or the shape of a water fountain.

What are some applications of reflected parabolas in science and technology?

Reflected parabolas have many applications in science and technology. In physics, they are used to model the path of projectiles and the motion of objects in free-fall. In engineering, they are used in the design of satellite dishes, parabolic mirrors, and other optical devices. In mathematics, they are used to solve optimization problems and to model real-world phenomena. They are also used in computer graphics to create realistic 3D images and animations.

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