31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola with focus at (3, 0)

Verified step by step guidance

1. Recall that a parabola with its vertex at the origin and focus on the x-axis has the standard form equation \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus.
2. Identify the value of \(p\) from the given focus. Since the focus is at \((3, 0)\), the distance \(p\) is 3.
3. Substitute \(p = 3\) into the standard form equation to get \(y^2 = 4 \times 3 \times x\).
4. Simplify the equation to \(y^2 = 12x\), which represents the parabola with vertex at the origin and focus at \((3, 0)\).
5. Verify the orientation: since the focus is on the positive x-axis, the parabola opens to the right, consistent with the equation \(y^2 = 4px\) where \(p > 0\).

Key Concepts

Definition of a Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Understanding this geometric definition is essential to derive the equation of a parabola given its focus and vertex.

Standard Form of a Parabola with Vertex at the Origin

When the vertex is at the origin, a parabola with a horizontal axis of symmetry has the form x² = 4py (vertical) or y² = 4px (horizontal). The parameter p represents the distance from the vertex to the focus, which helps in writing the equation.

Relationship Between Focus and Parameter p

The distance p is the distance from the vertex to the focus along the axis of symmetry. For a focus at (3, 0), p = 3, indicating the parabola opens horizontally. This value is used directly in the standard form equation y² = 4px.