The three types of conic sections are the hyperbola, the parabola, and the ellipse. The circle is type of ellipse, and is sometimes considered to be a fourth type of conic section. If the plane is parallel to the generating line, the conic section is a parabola.

Sal manipulates the equation x^2+y^2-3x+4y=4 in order to find that it represents a circle, and the equation 2x^2+y+12x+16=0 in order to find it represents a parabola.

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix).

1 : a plane curve generated by a point moving so that its distance from a fixed point is equal to its distance from a fixed line : the intersection of a right circular cone with a plane parallel to an element of the cone. 2 : something bowl-shaped (such as an antenna or microphone reflector)

A parabola is the set of points in a plane that are the same distance from a given point and a given line in that plane. The given point is called the focus, and the line is called the directrix. The equation of a parabola can be written in two basic forms: Form 1: y = a( x – h) 2 + k.

All parabolas are vaguely “U” shaped and they will have a highest or lowest point that is called the vertex. Every parabola has an axis of symmetry and, as the graph shows, the graph to either side of the axis of symmetry is a mirror image of the other side.

If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The y -intercept is the point at which the parabola crosses the y -axis. The x -intercepts are the points at which the parabola crosses the x -axis.

The graph of a quadratic equation in two variables (y = ax2 + bx + c ) is called a parabola. We say that the first parabola opens upwards (is a U shape) and the second parabola opens downwards (is an upside down U shape). In order to graph a parabola we need to find its intercepts, vertex, and which way it opens.

A hyperbola is the set of all points P in the plane such that the difference between the distances from P to two fixed points is a given constant. Each of the fixed points is a focus . (The plural is foci.) The center of a hyperbola is the midpoint of the line segment joining its foci.

Like hyperbolas centered at the origin, hyperbolas centered at a point (h,k) have vertices, co-vertices, and foci that are related by the equation c2=a2+b2 c 2 = a 2 + b 2 .

A circle is a special case of an ellipse because it is an ellipse where the diameter in both the x and y direction are the same.

A circle has only one radius—the distance from the center to any point is the same. To change our circle into an ellipse, we will have to stretch or squeeze the circle so that the distances are no longer the same.