The given function is:

$f(x) = |x + 1| - 7$

This is an absolute value function of the form $f(x) = |x - h| + k$, where the vertex of the function is located at $(h, k)$.

Here, we have $x + 1$ inside the absolute value, which means the expression is in the form $|x - (-1)| - 7$.

Thus, the vertex is at $(-1, -7)$.

The vertex of the function is $(-1, -7)$.

Would you like a detailed explanation of how to find the vertex of absolute value functions?

Here are some related questions:

1. How does the graph of an absolute value function look?
2. What happens to the vertex if the coefficient of the $x$-term changes?
3. How do you transform the graph of $f(x) = |x|$ by changing the constants?
4. What is the impact of adding a constant inside the absolute value?
5. How would you solve for the vertex of a quadratic function?

Tip: To graph absolute value functions, focus on identifying the vertex and understanding the "V" shape formed by the graph.