The given function is $y = -|x + 7| - 1$. Let's analyze the vertex, domain, and range:

1. Vertex:

The standard form of an absolute value function is $y = a |x - h| + k$, where $(h, k)$ is the vertex. Here:

• The function is $y = -|x + 7| - 1$, which can be rewritten as $y = -|x - (-7)| - 1$.
• Thus, the vertex occurs at $x = -7$ and $y = -1$.
• Vertex: $(-7, -1)$.

2. Domain:

• The absolute value function is defined for all real values of $x$. So, the domain is $(-\infty, \infty)$.
• Domain: $(-\infty, \infty)$.

3. Range:

• The function has a negative coefficient $-|x + 7|$, meaning it opens downwards.
• The vertex is at $y = -1$, which is the maximum value.
• Therefore, the range is $(-\infty, -1]$.
• Range: $(-\infty, -1]$.

Summary:

• Vertex: $(-7, -1)$.
• Domain: $(-\infty, \infty)$.
• Range: $(-\infty, -1]$.

Do you want more details or have any questions?

Related Questions:

1. How do we find the vertex of a general absolute value function?
2. What happens to the graph of $y = |x|$ when we introduce a negative coefficient?
3. How do transformations affect the domain of an absolute value function?
4. Can the range of an absolute value function be affected by horizontal shifts?
5. What is the effect of adding or subtracting constants inside the absolute value?

Tip:

Remember, the domain of an absolute value function is always all real numbers, but the range depends on vertical shifts and whether the function opens upwards or downwards.