Use graphing technology to find the range of the function f, of, x, equals, minus, x, squared, minus, 4, x, minus, 4, .f(x)=−x 
2
 −4x−4

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.

Use graphing technology to find the range of the function f(x) = −x^2 − 4x − 4.

Learn how to find the range of the quadratic function f(x) = −x^2 − 4x − 4 using graphing technology. This step-by-step solution covers concepts like vertex form, the quadratic equation, and determining the maximum value of a downward-opening parabola. Suitable for students in Grades 9-11.

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