Equation of a Parabola with a Specific Vertex and Axis

When dealing with geometric shapes like the parabola, understanding how to derive its equation based on given conditions is crucial. A parabola is a conic section formed by the intersection of a double-napped right circular cone with a plane parallel to the side of the cone. This article will focus on finding the equation of a parabola given a specific vertex and axis, as well as passing through a certain point. The process involves both theoretical and computational techniques to achieve the desired result.

Vertex and Axis of a Parabola

A parabola can be described in several ways, but in this context, we consider a parabola with its vertex at the origin and its axis along the x-axis. The standard form of such a parabola is given by the equation:

y^2 4ax

Here, the vertex of the parabola is at the origin (0, 0), and the axis is the x-axis. To find the specific equation, we need to determine the value of (a).

Deriving the Equation of the Parabola

To find the equation of the parabola that passes through the point (-3, 6), we substitute these coordinates into the general equation:

6^2 4a(-3)

Let's solve for (a):

36 -12a
a -3

Thus, the required equation of the parabola is:

y^2 12x

General Form of Parabolas with Different Orientations

While the above form is specific to a parabola with a vertex at the origin and axis along the x-axis, other forms exist for various orientations. These include:

x^2 4ay - with the vertex at the origin and axis along the y-axis.

y^2 -4ax - with the vertex at the origin and turning left.

x^2 -4ay - with the vertex at the origin and axis along the y-axis and turning left.

Alternative Approach: Deriving the General Equation

Another approach to finding the equation of the parabola involves starting with the general form:

y ax^2 bx c

Given that the vertex is at the origin (0, 0), the constant term (c) is 0. The parabola passes through the point (-3, 6), so we substitute these values into the equation:

6 9a - 3b
a frac{-3 cdot b}{9} frac{b - 2}{3}

For the parabola to open towards the left, (a) must be negative. A simple choice could be (b -5), which gives (a -1). Thus, the equation becomes:

y -frac{1}{3}x^2 - frac{5}{3}x - 2

Conclusion

In summary, the equation of a parabola with a vertex at the origin (0, 0) and axis along the x-axis, passing through the point (-3, 6), is derived to be:

y^2 12x

The process of deriving this equation involves a step-by-step algebraic manipulation based on the given conditions. Understanding these steps is essential in solving similar problems and in the broader context of conic sections.

Keywords: parabola, vertex, axis, equation, google seo