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

Explanation:

Step1: Recall the vertex - form of a parabola

The vertex - form of a parabola is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. Given the vertex $V(3,6)$, we have $h = 3$ and $k = 6$. So the equation of the parabola is $y=a(x - 3)^2+6$.

Step2: Find the value of $a$

Since the parabola passes through the point $(0,0)$, we substitute $x = 0$ and $y = 0$ into the equation $y=a(x - 3)^2+6$. [ \begin{align*} 0&=a(0 - 3)^2+6\ 0&=9a+6\

• 6&=9a\ a&=-\frac{2}{3} \end{align*} ]

Step3: Write the standard equation

Substitute $a=-\frac{2}{3}$ back into the vertex - form equation. The standard equation of the parabola is $y=-\frac{2}{3}(x - 3)^2+6$. Expanding it: [ \begin{align*} y&=-\frac{2}{3}(x^{2}-6x + 9)+6\ y&=-\frac{2}{3}x^{2}+4x-6 + 6\ y&=-\frac{2}{3}x^{2}+4x \end{align*} ]

Answer:

$y=-\frac{2}{3}x^{2}+4x$