To write the equation of a parabola, we can use the vertex form of a parabola's equation:

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

Where:

• $(h, k)$ is the vertex of the parabola.
• $a$ is a constant that determines the "width" and direction of the parabola.

Given:

• The vertex is $(2, -3)$, so $h = 2$ and $k = -3$.
• A point on the parabola is $(3, 1)$, which will allow us to solve for $a$.

Step 1: Substitute the vertex into the vertex form equation:

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

Step 2: Substitute the point $(3, 1)$ into the equation to find $a$:

When $x = 3$, $y = 1$: $1 = a(3 - 2)^2 - 3$ $1 = a(1)^2 - 3$ $1 = a - 3$ $a = 4$

Step 3: Substitute $a = 4$ into the vertex form equation:

$y = 4(x - 2)^2 - 3$

Thus, the equation of the parabola is: $y = 4(x - 2)^2 - 3$

Would you like more details or have any questions?

Here are 5 related questions to extend this topic:

1. How do you convert a parabola from vertex form to standard form?
2. What does the value of $a$ signify in the vertex form of a parabola?
3. How would the equation change if the parabola opened downwards?
4. What are the axis of symmetry and the direction of the parabola in this case?
5. How do you find the focus and directrix of a parabola given its equation?

Tip: The vertex form of a parabola's equation is useful because it immediately gives the vertex, making it easier to graph and understand the parabola's properties.