Questions: Find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation. Directrix the line y = -3/2 ; vertex at (0,0)

Solution Steps

The vertex of the parabola is at the origin, \((0,0)\), and the directrix is the line \(y = -\frac{3}{2}\). Since the directrix is horizontal, the parabola opens upwards. The distance from the vertex to the directrix is \(\frac{3}{2}\), so the focus is at \((0, \frac{3}{2})\).

The standard form of a parabola that opens upwards is: \[ y = \frac{1}{4p}x^2 \] where \(p\) is the distance from the vertex to the focus. Here, \(p = \frac{3}{2}\), so: \[ y = \frac{1}{4 \times \frac{3}{2}}x^2 = \frac{1}{6}x^2 \]

The latus rectum is a line segment perpendicular to the axis of symmetry of the parabola and passes through the focus. Its length is \(4p\).

For this parabola, \(4p = 4 \times \frac{3}{2} = 6\).

The focus is at \((0, \frac{3}{2})\), so the endpoints of the latus rectum are: \[ \left(-3, \frac{3}{2}\right) \quad \text{and} \quad \left(3, \frac{3}{2}\right) \]

Final Answer