Question

write an equation of the parabola shown. an equation of the parabola is

Explanation:

Step1: Determine the standard form

The standard form of a parabola that opens to the right is \((y - k)^2=4p(x - h)\), where \((h,k)\) is the vertex.

Step2: Identify the vertex

From the graph, the vertex \((h,k)=(0,0)\).

Step3: Find the value of \(p\)

The directrix of a parabola \((y - k)^2 = 4p(x - h)\) is \(x=h - p\). Given the directrix \(x=\frac{2}{5}\), and \(h = 0\), then \(0-p=\frac{2}{5}\), so \(p=-\frac{2}{5}\).

Step4: Substitute into the standard form

Substitute \(h = 0\), \(k = 0\), and \(p=-\frac{2}{5}\) into \((y - k)^2=4p(x - h)\). We get \(y^{2}=4\times(-\frac{2}{5})x\), which simplifies to \(y^{2}=-\frac{8}{5}x\).

Answer:

\(y^{2}=-\frac{8}{5}x\)