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

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Identify the vertex of the parabola, which is given as (0, 2). This is the highest point on the parabola since it opens downward.

Note the directrix line, which is horizontal and given by the equation \(y = 4\). The parabola opens downward because the vertex is below the directrix.

Find the focus of the parabola. The focus lies the same distance from the vertex as the directrix but on the opposite side. Calculate the distance \(p\) between the vertex and the directrix: \(p = 4 - 2 = 2\). Since the parabola opens downward, the focus is at \((0, 2 - 2) = (0, 0)\).

Use the standard form of a vertical parabola with vertex \((h, k)\) and focus distance \(p\(: \[(x - h)^2 = 4p(y - k)\]. Here, \)h = 0\), \(k = 2\), and \(p = -2\) (negative because it opens downward). Substitute these values to get the equation.

Write the final equation of the parabola as \[x^2 = 4(-2)(y - 2)\] or simplified to \[x^2 = -8(y - 2)\].

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Parabola Definition and Properties

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. The vertex is the midpoint between the focus and directrix. The parabola opens away from the directrix, and its shape depends on the distance between the vertex and the focus.

Equation of a Parabola with Vertical Axis

For a parabola with a vertical axis of symmetry, the standard form is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus. If p is positive, the parabola opens upward; if negative, it opens downward.

Using the Directrix to Find the Parabola Equation

The directrix is a line equidistant from the vertex as the focus but in the opposite direction. Knowing the directrix's equation helps determine p and the vertex's position, which are essential to write the parabola's equation accurately.