Determining the Equation of a Parabola Given its Vertex and Focus

When dealing with parabolas in geometry and algebra, it's essential to understand the properties of the vertex and focus to derive the equation of the parabola. In this article, we will explore the equation of a parabola whose vertex is the point (34) and focus is the point (12).

Understanding the Parabola's Components

A parabola is a conic section that can be defined as a set of points in a plane that are equidistant from a fixed point (the focus) and a fixed straight line (the directrix). The vertex of a parabola is the point where the parabola changes direction, and the distance between the vertex and the focus determines the parabola's opening.

Calculating the Distance Between the Vertex and the Focus

The first step is to determine the distance between the vertex (34) and the focus (12). The distance ( a ) is calculated using the distance formula:

[ a sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2} ]

Plugging in the coordinates of the vertex and the focus:

[ a sqrt{(12 - 34)^2 (2 - 4)^2} ]

Simplifying further:

[ a sqrt{(-22)^2 (-2)^2} ] [ a sqrt{484 4} ] [ a sqrt{488} ]

Therefore, the value of ( a ) is ( sqrt{488} ), which simplifies to ( sqrt{16 times 31} ) or ( 4sqrt{31} ).

Deriving the Equation of the Parabola

The equation of a parabola with a vertical axis of symmetry can be written as:

[ (y - k)^2 4p(x - h) ]

Where ( (h, k) ) is the vertex of the parabola, and ( p ) is the distance from the vertex to the focus. In this case, the vertex is (34) and the focus is (12), so ( h 34 ), ( k 4 ), and ( p a 4sqrt{31} ).

Substituting these values into the equation:

[ (y - 4)^2 4(4sqrt{31})(x - 34) ]

Simplifying the right side:

[ (y - 4)^2 16sqrt{31}(x - 34) ]

This is the equation of the parabola. It describes how the y-coordinates of points on the parabola vary with the x-coordinates, with the vertex at (34) and the focus at (12).

Graphical Representation and Analysis

To better visualize this parabola, one can plot the points and sketch the curve. The vertex (34) is the lowest point of the parabola, and it opens to the right since the focus is to the left of the vertex.

The distance ( a 4sqrt{31} ) determines how far the parabola opens. This distance helps in understanding the overall shape and positioning of the parabola in the coordinate plane.

Conclusion and Further Exploration

Understanding the relationship between the vertex and the focus is crucial for deriving the equation of a parabola. In this case, we have successfully determined the equation of a parabola whose vertex is (34) and focus is (12).

For further exploration, you can try deriving the equation of a parabola with different vertex and focus points or experiment with different algebraic manipulations of the parabolic equation. Additionally, you can use graphing software to visualize and understand the graphical representation of such equations more clearly.

By mastering these concepts, you will be well-equipped to handle more complex problems in geometry and algebra.