Sorry for being unclear. I do not mean a solid of revolution. I mean taking a parabola which lies on the xy plane and rotating it (moving all of its points) by some amount of degrees around an axis (in my case a line which goes through points (1,0) and (0,1). If you look at the picture it represents a rotated parabola which "had" its points of intersection with x and y axes fixed.HallsofIvy said:When you say "a parabola in the xy-plane" which parabola do you mean? If you rotate a parabola around any other line but its axis, the result is NOT a paraboloid.
The equation for a tilted parabola in 3D is z = ax2 + by2 + cxy + dx + ey + f, where a, b, and c determine the shape and orientation of the parabola and d, e, and f determine its position in 3D space.
To graph a tilted parabola in 3D, you can plot points on a 3D coordinate system using the equation z = ax2 + by2 + cxy + dx + ey + f. Alternatively, you can use a software program or graphing calculator to plot the parabola for you.
The c term in the equation for a tilted parabola in 3D determines the level of asymmetry in the parabola. A positive value for c will result in a parabola that opens to the right, while a negative value will result in a parabola that opens to the left. A value of 0 for c will result in a symmetrical parabola.
Yes, a tilted parabola in 3D can have a vertex at the origin if the d and e terms in the equation are both equal to 0. In this case, the parabola will open either up or down, depending on the values of a and b.
A tilted parabola in 3D is different from a regular parabola in 2D in that it has an additional variable, z, which represents the third dimension. This allows for more complex and asymmetrical parabolas that cannot be represented in 2D. Additionally, the c term in the equation for a tilted parabola in 3D is not present in the equation for a regular parabola in 2D, as the latter is always symmetrical along the y-axis.