To write the equation of a parabola in vertex form, the general formula is:

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

Where:

• $(h, k)$ is the vertex of the parabola.
• $a$ is a constant that controls the "width" and direction of the parabola.
• $(x, y)$ is any other point on the parabola.

Step 1: Substitute the vertex $(h, k) = (3, 3)$ into the vertex form:

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

Step 2: Use the point $(2, -3)$ to solve for $a$.

Substitute $(x, y) = (2, -3)$ into the equation: $-3 = a(2 - 3)^2 + 3$ $-3 = a(-1)^2 + 3$ $-3 = a(1) + 3$ $-3 - 3 = a$ $a = -6$

Step 3: Write the final equation of the parabola:

$y = -6(x - 3)^2 + 3$

This is the equation of the parabola in vertex form.

Do you want any additional details, or have any questions about this?

Here are 5 related questions for deeper exploration:

1. How does the value of $a$ affect the shape of the parabola?
2. What is the axis of symmetry for this parabola?
3. How can you convert this equation to standard form?
4. What would happen if the vertex was shifted to a different point?
5. How can we find the focus and directrix of this parabola?

Tip: The sign of $a$ in a parabola's equation tells you whether it opens upwards ($a > 0$) or downwards ($a < 0$).