The function given is $f(x) = -|x + 1| - 2$.

Analysis:

1. Absolute Value: The term $|x + 1|$ represents the absolute value of $x + 1$. This means the expression inside the absolute value is always non-negative.

2. Negative Sign: The function has a negative sign in front of the absolute value, $-|x + 1|$, which reflects the graph of $|x + 1|$ downward.

3. Vertical Shift: The $-2$ at the end shifts the entire graph downward by 2 units.

Key Points:

• Vertex: The vertex of the function occurs where the expression inside the absolute value is zero, i.e., $x + 1 = 0$, so $x = -1$. The corresponding $y$-value is $f(-1) = -|0| - 2 = -2$. Thus, the vertex is at $(-1, -2)$.

• Shape of the Graph: Since the function involves the negative of an absolute value, the graph is a downward-pointing "V" shape.

• Behavior:

  • For $x > -1$, $f(x) = -(x + 1) - 2$.
  • For $x < -1$, $f(x) = (x + 1) - 2$.

Graph of the Function:

The graph has a vertex at $(-1, -2)$ and opens downward.

Would you like a detailed explanation on any part of this, or do you have further questions?

Follow-up Questions:

1. How does the graph of $|x + 1|$ compare to the graph of $-|x + 1|$?
2. What happens to the function $f(x)$ if we change $-2$ to $+2$?
3. Can you find the x-intercepts of the function $f(x)$?
4. How does the graph of $f(x)$ shift if the equation is $f(x) = -|x - 1| - 2$?
5. What is the domain and range of the function $f(x) = -|x + 1| - 2$?

Tip:

When working with absolute value functions, remember that the graph typically forms a "V" shape, and modifications such as a negative sign or vertical shifts will affect its orientation and position on the coordinate plane.