31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

Verified step by step guidance

1. Identify the orientation of the parabola. Since the parabola opens to the right and the vertex is at the origin, the standard form of the equation is \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus.
2. Determine the value of \(p\) by using the coordinates of the focus. The focus is at \((-1, 0)\), so the distance from the vertex \((0,0)\) to the focus is \(|p| = 1\). Since the parabola opens to the right, \(p\) should be positive, but here the focus is to the left, so \(p = -1\).
3. Write the equation of the parabola using the value of \(p\). Substitute \(p = -1\) into the standard form to get \(y^2 = 4(-1)x\), which simplifies to \(y^2 = -4x\).
4. Verify the equation by checking the directrix. The directrix is a vertical line given by \(x = -p\), so with \(p = -1\), the directrix is \(x = 1\). However, the problem states the directrix is \(x = -2\), so re-examine the value of \(p\) considering the directrix.
5. Calculate \(p\) using the directrix \(x = -2\) and the vertex at \((0,0)\). The distance from the vertex to the directrix is \(|p| = 2\), and since the parabola opens to the right, \(p = 2\). Therefore, the correct equation is \(y^2 = 4(2)x\), or \(y^2 = 8x\).

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. This geometric definition helps derive the equation of the parabola by setting the distance from any point on the curve to the focus equal to its distance to the directrix.

Standard Form of Parabola Equations

Parabolas with vertices at the origin have standard forms: y² = 4px for horizontal parabolas and x² = 4py for vertical parabolas. The parameter p represents the distance from the vertex to the focus (or directrix), determining the parabola's width and direction of opening.

Using Focus and Directrix to Find Equation

Given the focus and directrix, the value of p is half the distance between them. The vertex lies midway between the focus and directrix. Using these, substitute p into the standard form to write the parabola's equation, ensuring the parabola opens toward the focus.