How to Find the Equation of a Parabola with a Given Focus and Vertex

The equation of a parabola can be determined when both the vertex and focus are known. In many geometric problems, the parabola often has its vertex at the origin, and the focus at a specific point, such as (7, 0). In this tutorial, we explore the steps to find the equation of a parabola with its vertex at the origin and a focus at (7, 0).

Understanding the Properties of a Parabola

A parabola is a curve where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix). The vertex of the parabola is the point where the parabola changes direction, and it lies exactly between the focus and the directrix. For a standard parabola that opens to the right, the vertex is at the origin (0, 0)

Equation of the Parabola

The standard form of the equation of a parabola that opens to the right is given by:

y2 4px

where p is the distance from the vertex to the focus. When the vertex is at the origin (0, 0) and the focus is at (7, 0), p is 7 units.

Deriving the Equation

Given the focus at (7, 0), p 7. Substituting this value into the standard form gives:

y2 4*7*x

This simplifies to:

y2 28x

Therefore, the equation of the parabola is:

y2 28x

Verification with Wikipedia

According to Wikipedia, a parabola is defined by its directrix d and focus F. The distance from the focus to the directrix is denoted as p. For a parabola with its vertex at the origin and focus at (7, 0), the equation can be derived as:

(y - 0)2 4*7*(x - 0)

This simplifies directly to the same equation:

y2 28x

Graphical Approach

A graphical solution involves constructing a basic parabola with the focus at (-0.25). If the y-value of the focus is multiplied by 7 (the distance from the vertex to the focus), the x-values of the points on the parabola (0.5 and -0.5) will be scaled by a factor of 28. This yields the equation:

28y x2

By reversing the roles of x and y, we obtain:

28x y2

This confirms the derived equation as:

y2 28x

Conclusion

The equation of a parabola with a vertex at the origin and a focus at (7, 0) is given by y2 28x. This formula uses the distance p 7 from the vertex to the focus, and the standard form of the equation for a parabola that opens to the right. Understanding these properties and the process of derivation helps in solving similar problems.

Key Points

The vertex of the parabola is at the origin (0, 0). The focus is at (7, 0). The value of p is 7 units. The equation of the parabola is y2 28x.

By following this method, you can derive the equation of a parabola given its vertex and focus, a fundamental concept in analytical geometry and calculus.