To find the equation of a quadratic function, we start with the general form of a parabola:

$y = a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola. Given that the vertex is at $(3, 9)$, the equation becomes:

$y = a(x - 3)^2 + 9$

Next, we use the fact that the parabola passes through the origin, $(0, 0)$. Substituting $x = 0$ and $y = 0$ into the equation gives:

$0 = a(0 - 3)^2 + 9$

$0 = 9a + 9$

Solving for $a$, we get:

$9a = -9 \implies a = -1$

Thus, the equation of the quadratic function is:

$y = -(x - 3)^2 + 9$

Expanding this equation, we get:

$y = -(x^2 - 6x + 9) + 9$

$y = -x^2 + 6x$

Therefore, the equation of the quadratic function is:

$y = -x^2 + 6x$

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. How would the equation change if the parabola opened upward?
2. What is the axis of symmetry for this quadratic function?
3. How can you find the x-intercepts of the function $y = -x^2 + 6x$?
4. How does the value of $a$ affect the width of the parabola?
5. How would you find the y-intercept of a quadratic function given its equation?

Tip: The vertex form of a quadratic equation makes it easy to identify the vertex and the direction in which the parabola opens (based on the sign of $a$).