A parabola is a smooth curve that has the formula: y = ax^2 + bx + c;
where “a” is any real number, and “x” represents the position of each point on the graph. If the value of “a”, or coefficient of “x”, is less than zero, then it will be shown as a vertical line (similar to x= 5), and if it’s greater than zero, then it’ll show itself as a horizontal line (similar to x= 5). This article will represent the equation of parabola. A graph with an even quadratic function must have at least one root, but may have multiple roots, including real and complex solutions. For example: y=/4 would represent y = 0, y = 4.
 Parabola is also an optical curve that forms an image in a conic surface with the light passing through one focus (the simplest case). A parabolic mirror can be used to form a real, inverted image of a single incoming object at its principal focus. The reflective element in a flashlight is a parabolic shape which reflects light from its filament into a smooth pool of light on the nearby flat wall of its reflector.
 In addition, And if “x” represents length and “y” represents width, then the difference between the maximum and minimum values is equal to 2a inches. In this same sense, when one substitutes length for x and width for y, it tells one that the sum of the maximum and minimum values is equal to 2a inches.
 When one substitutes length for x and width for y, it tells one that the sum of the maximum and minimum values is equal to 2a inches. Because parabolas are symmetrical, if “x” represents height, then y will represent depth; making the difference between height (maximum) minus depth (minimum) also equals 2a.
 Similarly, when “y” represents height, “x” will represent depth; again making the difference between y (maximum) minus x (minimum), equals 2a.
 The equation of parabola can be expressed as:
The parabola equation is y=ax^2+bx+c, where “a” is any real number, and “x” represents the position of each point on the graph.

Properties of a parabola?
A parabola cannot be an odd function because it has an even degree 2 in its formula. This means that it will always have at least one root (solution). However, there may be more than one solution to the equation depending on the coefficient (a) value in the exponent (x^2). Also, if “A”, or coefficient (a), equals zero, then this indicates that there is no real solution, meaning all possible answers will either be imaginary or no solution.

How can one identify a parabola?
Finding if a graph is a parabola can be done by finding the value of “a” in the formula y=ax^2+bx+c. If it’s less than zero, then this will indicate that it is a vertical line (similar to x=5). In the opposite sense, if it’s greater than zero, it’ll show itself as a horizontal line (similar to x=5).

The equation of ellipsecan be expressed as:
y^2/ a^2 = x^2/ b^2, where “a” is the radius of the vertical axis and “b” is the radius of the horizontal axis.

Properties of ellipses:
An ellipse has at least one real solution, but may have no solutions depending on what value coefficient (a) or (b) takes. In addition, if “A”, coefficient (a), equals zero, this indicates that there will be imaginary solutions because it cancels out y in the equation.

How can one identify ellipses?
One would find if a graph is an ellipse by finding the value of “a” or coefficient (a) in equation y^2/ a^2 = x^2/ b^2. If it’s greater than zero, this will show itself as an ellipse; therefore, what to look for is its horizontal axis which goes through the origin point (0,0). If it’s less than zero, this will indicate that it is a hyperbola. In the opposite sense, if “a” equals zero, this will indicate that it is a parabola, same as (x=5), which tells one that it’s a vertical line.
With the additional help of the Cuemath website, one can easily understand challenging concepts like this.