Question

which is the standard form of the equation of the parabola shown in the graph?

Explanation:

Step1: Recall the standard - form of a parabola

The standard form of a parabola with a vertical axis of symmetry is \((x - h)^2=4p(y - k)\), where \((h,k)\) is the vertex and \(p\) is the distance between the vertex and the focus or the vertex and the directrix.

Step2: Identify the vertex of the parabola

From the graph, the vertex of the parabola is \((0,0)\), so \(h = 0\) and \(k = 0\). The equation becomes \(x^{2}=4p y\).

Step3: Determine the value of \(p\)

The directrix is \(y = 4\) and the vertex is \((0,0)\). The distance \(p\) from the vertex to the directrix for a parabola opening downwards is negative. The distance from the vertex \((0,0)\) to the directrix \(y = 4\) is \(4\), so \(p=- 4\).

Step4: Substitute \(p\) into the equation

Substitute \(p=-4\) into \(x^{2}=4p y\), we get \(x^{2}=4\times(-4)y\), which simplifies to \(x^{2}=-16y\).

Answer:

\(x^{2}=-16y\)