The function is given as $f(x) = -x^2 - 4x - 4$, which is a quadratic function in the form $f(x) = ax^2 + bx + c$, where $a = -1$, $b = -4$, and $c = -4$.

To determine the range, we will follow these steps:

Step 1: Identify the Vertex

Since this is a quadratic function, its graph is a parabola. The vertex form of a quadratic equation is $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex. For a downward-opening parabola (since $a = -1$), the maximum value of the function occurs at the vertex.

To find the x-coordinate of the vertex, use the formula: $h = -\frac{b}{2a}$ Substitute $a = -1$ and $b = -4$: $h = -\frac{-4}{2(-1)} = \frac{4}{-2} = -2$

Step 2: Find the Maximum Value (y-coordinate of the vertex)

Now, substitute $h = -2$ back into the original function to find the maximum value (i.e., the y-coordinate of the vertex): $f(-2) = -(-2)^2 - 4(-2) - 4$ $f(-2) = -4 + 8 - 4 = 0$ Thus, the vertex is $(-2, 0)$.

Step 3: Determine the Range

Since the parabola opens downward, the function's maximum value is $0$ at $x = -2$. The function decreases without bound as $x$ moves away from the vertex in either direction.

Therefore, the range of the function is: $(-\infty, 0]$

Would you like more details or have any questions?

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Here are 5 related questions:

1. How do you convert a quadratic function to vertex form?
2. What does the sign of the coefficient $a$ in a quadratic function tell you about the direction of the parabola?
3. How do you find the axis of symmetry for a quadratic function?
4. How do you determine if a quadratic function has a maximum or a minimum value?
5. What is the domain of any quadratic function?

Tip: The vertex of a quadratic function represents either the maximum or minimum value, depending on whether the parabola opens upward or downward.