Student Question

# Determine at sight the direction in which the curve opens. Locate the vertex, focus, and ends of the latus rectum: x^2+8y=0

A parabola opens upwards if its general equation is of the form x^2 = 4ay and sidewards if its general equation is of the form y^2 = 4ay.

For y^2 = 4ax, the vertex is (0,0). For any other vertex (x0, y0) the equation is of the form (y - y0)^2 = 4a(x - x0). So you can determine the vertex by seeing where the parabola intersects the axis. The focus is a point inside the parabola at a distance a from the vertex and lying on the axis.

The ends of the latus rectum are points of a chord drawn through the focus, parallel to the directrix which end on the parabola. For a parabola opening upward, they would have the same y-coordinate as the focus and the x-coordinate would be +2a and -2a.

For x^2 + 8y = 0 => x^2 = -8y=> a = -2

The parabola opens in the downwards direction. The vertex is (0,0). The focus is (0,-2) and the ends of the latus rectum are (4, -2) and (-4, -2).