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How To Find the Focus of a Parabola: Your Easy Guide

If you want to know how to find the focus of a parabola, you first need to define what a parabola is. A parabola is a curved figure where any point on the curve is equal distance from a fixed point (called the focus) and a fixed straight line (called the directrix).

how to find the focus of a parabola: Diagram of a parabola

Let’s identify the parts of a parabolic function. In the graph above, you see a given line that intersects the directrix at a 90-degree angle. This straight line is called the axis of symmetry. The point that is marked C, signifying where the parabola opens, is called the vertex. The vertex is always midway between the focus and directrix of a parabola.

The Equation of a Parabola

how to find the focus of a parabola: Basic representation of a parabola

The above graph is a basic representation of a parabola where the coordinates of the vertex are (0,0). When you draw the axis of symmetry through the parabola's vertex, you see that this vertical line perfectly matches up with the y-axis of the graph. This parabola is represented by equation how to find the focus of a parabola: Equation representing a parabola

how to find the focus of a parabola: Graph showing a parabola rotated 90 degrees to the right

If this parabola was rotated 90 degrees to the right, the fixed line representing the axis of symmetry would be situated along the x-axis. The equation of the parabola is now how to find the focus of a parabola: Equation of a parabola.

The standard form of the equation of a parabola, where the conic shape of the parabola is formed along the y-axis, is how to find the focus of a parabola: standard form of the equation of a parabola. Coefficients h and k represent the points of the vertex. Coefficient p represents the distance from the vertex to the focus, which is equal to the distance from the vertex to the directrix.

How To Find the Focus of a Parabola

In order to find the focus of a parabola, you must know that the equation of a parabola in a vertex form is y=a(x−h)2+k where a represents the slope of the equation. From the formula, we can see that the coordinates for the focus of the parabola is (h, k+1/4a).

So now, let's solve for the focus of the parabola below:

how to find the focus of a parabola: Graph of a parabola

When we use the above coordinates, the equation of the parabola above is Equation of a parabola. We can plug in Value of a in the equation for a and (0,0) for the vertex. Now let's solve for the y-coordinate of the focus:

Formula to get the y-coordinate

Formula to get the y-coordinate

Formula to get the y-coordinate

Formula to get the y-coordinate

We've determined that the points of the focus are (0,2).

Let’s Review How To Find the Focus of a Parabola

In order to determine how to find the focus of a parabola, you must identify the vertex and plug it into the equation of a parabola. Understanding different properties of a parabola, like the axis of symmetry, directrix, and reflectors, allows you to expand upon your basic understanding of graphing geometric equations.

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