How do you prove that an equation is a parabola?

One way to show that the general quationy=ax2+bx+c is a parabola is to show that any such equation (with a≠0) is transformed by a unique homothety-translation into y=x2 and verify that its focus is (0,d) and directrix is y=−d as in the article parabola by showing that x2+(y−d)2=(y+d)2 where d=1/4.

How is a parabola defined geometrically?

Geometric Definition Definition: A parabola is the set of points in the plane that are equidistant from a point (the focus) and a line (the directrix.) Calculate its distance to the focus, its distance to the directrix, set those equal, and simplify.

What is algebra and analytic geometry?

In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables.

What differences are noticeable between analytic geometry as developed by Fermat and by Descartes?

The key difference between Fermat’s and Descartes’ treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves.

How many points define a parabola?

Two degrees of freedom are used to specify that point. Use two more to determine the line which is the directrix. You can do that with one more point if you take the directrix to be the line through that point perpendicular to the line joining the two points. Thus, you can use two points to define a parabola.

What are the properties of a parabola?

The parabola is symmetric about its axis. The axis is perpendicular to the directrix. The axis passes through the vertex and the focus. The tangent at vertex is parallel to the directrix.

Why is it called analytic geometry?

analytic geometry, also called coordinate geometry, mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry. The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations.

How do algebra and geometry work together to get geometric results within the coordinate plane?

Coordinate Geometry: One way that algebra and geometry can be related is through the use of equations in graphs. We can plot a set of points (x, y) according to an equation (for example, the line graph on the left!) to form a graph. That’s one way that algebra is related to geometry.

How do you find the points of a parabola?

Given y = ax2 + bx + c , we have to go through the following steps to find the points and shape of any parabola:

1. Label a, b, and c.
2. Decide the direction of the paraola:
3. Find the x-intercepts:
4. Find the y-intercept:
5. Find the vertex (h,k):
6. Plot the points and graph the parabola.