To find the equation of the parabola, we use the vertex form of a quadratic function:

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

Here, $(h, k)$ is the vertex, and $(x, y)$ is any point on the parabola.

Given:

• Vertex: $(3, -2)$, so $h = 3$ and $k = -2$.
• Point on the parabola: $(4, 3)$, so $x = 4$ and $y = 3$.

Substituting the vertex into the vertex form, we have:

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

Now, use the point $(4, 3)$ to solve for $a$. Plug in $x = 4$ and $y = 3$:

$3 = a(4 - 3)^2 - 2$

$3 = a(1)^2 - 2$

$3 = a - 2$

Solving for $a$:

$a = 5$

Thus, the equation of the parabola is:

$f(x) = 5(x - 3)^2 - 2$

Would you like more details or have any further questions?

Here are 5 related questions:

1. What is the general form of a quadratic equation?
2. How do you convert from vertex form to standard form?
3. How can you find the axis of symmetry for a parabola?
4. What role does the coefficient $a$ play in the shape of the parabola?
5. How can you determine if the parabola opens upwards or downwards?

Tip: The value of $a$ affects the "width" of the parabola. A larger absolute value of $a$ makes the parabola narrower, while a smaller absolute value makes it wider.