Question

find the standard equation of the parabola whose graph is shown.
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Explanation:

Step1: Determine the standard form of the parabola

Since the parabola opens upward and has its vertex at the origin \((0,0)\), the standard form is \(x^{2}=4py\).

Step2: Find the value of \(p\)

We know that the point \((3, p)\) lies on the parabola. Substitute \(x = 3\) and \(y=p\) into \(x^{2}=4py\).
So, \(3^{2}=4p\times p\), which simplifies to \(9 = 4p^{2}\), then \(p^{2}=\frac{9}{4}\), and \(p=\frac{3}{2}\) (since \(p>0\) as the parabola opens upward).

Step3: Write the equation of the parabola

Substitute \(p = \frac{3}{2}\) into \(x^{2}=4py\).
We get \(x^{2}=4\times\frac{3}{2}y\), which simplifies to \(x^{2}=6y\).

Answer:

\(x^{2}=6y\)