In such a reflection, the images of the points (1,0) and (0,1) are (0, -1) and (-1,0), respectively.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.
YesMonoxdifly said:\(\displaystyle (-x)^2-2(-x)-4(-y)-11=0\)?
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.
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).
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.
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.
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.