Standard Parabolas

You should already be familiar with parabolas from your previous studies in math, but we want to view them from a slightly different perspective. Let's start by looking at parabolas that are located at the origin. Then we will look at what happens if they are shifted to other locations in the \(xy\)-plane.

A parabola is a curve formed by all points that are an equal distance from a fixed point called a focus and a fixed line called a directrix.

Graph of a parabola opening upward with a vertical axis of symmetry passing through the vertex. The focus point is located on the axis of symmetry inside the curvature of the parabola. A horizontal line called the directrix is located below the parabola, and the distance between the directrix and vertex is the same as the distance between the focus and vertex. A point (x,y) is located on the curve of the parabola, and the distances from this point to the vertex is the equal to the distance from the point to the directrix.

The vertex of a parabola is the point located on the curve that is closest to both the focus and directrix. The graph of a parabola is symmetric across the line running through the vertex that is perpendicular to the diretrix, and we call this line the axis of symmetry.

Use the following graph to explore the shape and key property of parabolas. Select the Parabola option and then drag the blue and orange points to explore the graph's characteristics.

The equation of a parabola has two forms, depending on whether the parabola is oriented vertically or horizontally. If you remember from your earlier math classes, the graph of \(y = ax^2\) is oriented vertically, opening upwards if \(a > 0\) and downwards if \(a < 0\). Similarly, the graph of \(x = ay^2\) is oriented horizontally, opening to the right if \(a > 0\) and to the left if \(a < 0\).

The standard conic equations of a parabola with vertex at the origin have the following form.

Vertical Axis of Symmetry

Horizontal Axis of Symmetry

\[x^2 = 4py\]

The focus is located at the point \((0,p)\) and the directrix is the horizontal line \(y = -p\). The parabola will open up if \(p > 0\) or down if \(p < 0\). The length of the focal diameter is \(|4p|\).

Graph of a parabola opening to the right with a horizontal axis of symmetry passing through the vertex, which is located at the origin. The focus point is located on the axis of symmetry inside the curvature of the parabola and the vertical directrix is located to the left of the parabola. The distance between the focus and vertex and between the directrix and vertex are both labeled as p. The width of the parabola is measured by the focal diameter, which is parallel to the directrix and extends from one side of the parabola to the other side while passing through the focus.

\[y^2 = 4px\]

The focus is located at the point \((p,0)\) and the directrix is the vertical line \(x = -p\). The parabola will open right if \(p > 0\) or left if \(p < 0\). The length of the focal diameter is \(|4p|\).

The main difference in these two equations is whether the parabola is oriented horizontally or vertically. In both cases, the vertex is located at the origin. The focus and directrix are both the same distance, represented by \(p\), from the vertex. The parabola will always pass through the vertex while curving around the focus and away from the directrix. The focal diameter describes the width of the parabola measured through the focus, and this distance is equal to \(|4p|\).

The key characteristics of a parabola in conic form are the coordinates of its vertex and focus, the equations of its directrix and axis of symmetry, and the length of its focal diamter.

We need to be able to write the standard equation of a parabola given information about it, such as the location of its focus and directrix or its graph. We also need to be able to sketch a graph of a parabola given the equation and identity its key characteristics. Let's look at a few examples.