Given that the vertex of the quadratic function $y = f(x)$ is $(-4, 7)$, we can express the quadratic function in the vertex form:

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

We also know that the function passes through the point $(-1, -2)$. We can use this information to find the value of $a$.

Substitute $(-1, -2)$ into the equation:

$-2 = a(-1 + 4)^2 + 7$

Simplify inside the parentheses:

$-2 = a(3)^2 + 7$

$-2 = 9a + 7$

Solve for $a$:

$-2 - 7 = 9a$

$-9 = 9a$

$a = -1$

Now substitute $a$ back into the equation:

$f(x) = -1(x + 4)^2 + 7$

Thus, the quadratic function is:

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

Would you like more details or have any questions?

Here are some additional questions you might find useful:

1. How do you convert a quadratic function from standard form to vertex form?
2. What is the significance of the vertex in a quadratic function?
3. How do you determine the axis of symmetry for a quadratic function?
4. What are the steps to find the roots of a quadratic equation?
5. How does changing the coefficient $a$ in a quadratic function affect its graph?

Tip: Always double-check your work by substituting known points back into the equation to verify your solution.