Question

find the standard equation of the parabola shown in the figure.

Explanation:

Step1: Recall parabola vertex - form

The standard form of a parabola with vertex $(h,k)$ is $y=a(x - h)^2+k$. Here, the vertex $V(h,k)=(3,6)$, so the equation is $y=a(x - 3)^2+6$.

Step2: Find the value of $a$

Since the parabola passes through the point $(0,0)$, substitute $x = 0$ and $y = 0$ into the equation $y=a(x - 3)^2+6$. We get $0=a(0 - 3)^2+6$. Simplify it: $0 = 9a+6$. Then $9a=-6$, and $a=-\frac{2}{3}$.

Step3: Write the final equation

Substitute $a = -\frac{2}{3}$ back into the vertex - form equation. The standard equation of the parabola is $y=-\frac{2}{3}(x - 3)^2+6$.

Answer:

$y=-\frac{2}{3}(x - 3)^2+6$