The problem asks to write the equation of the parabola in vertex form, given a vertex $(4, 2)$ and another point $(3, -5)$.

The vertex form of a parabola is:

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

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

Step 1: Use the given vertex $(h, k) = (4, 2)$ to substitute into the vertex form equation:

$f(x) = a(x - 4)^2 + 2$

Step 2: Substitute the point $(3, -5)$ into the equation to solve for $a$.

Given that $f(3) = -5$, substitute $x = 3$ and $f(x) = -5$ into the equation:

$-5 = a(3 - 4)^2 + 2$

Simplify:

$-5 = a(1)^2 + 2$

$-5 = a + 2$

Step 3: Solve for $a$.

$a = -5 - 2 = -7$

Step 4: Write the final equation.

Now that we know $a = -7$, substitute it back into the equation:

$f(x) = -7(x - 4)^2 + 2$

This is the equation of the parabola in vertex form.

Would you like further details or clarifications?

Here are 5 related questions you might explore:

1. How does the value of $a$ affect the shape of the parabola?
2. How would you find the equation if given two points instead of the vertex?
3. Can you rewrite this equation in standard form?
4. What is the axis of symmetry of this parabola?
5. How can you find the focus and directrix of this parabola?

Tip: The sign of $a$ determines whether the parabola opens upwards (positive $a$) or downwards (negative $a$).