Use graphing technology to find the range of the function f, of, x, equals, minus, vertical bar, x, plus, 2, vertical bar, minus, 2, .f(x)=−∣x+2∣−2.

To find the range of the function \( f(x) = -|x + 2| - 2 \), we can analyze the behavior of the function step-by-step and use graphing tools to confirm the result.

### 1. **Understand the function structure**:
   - The absolute value function \( |x + 2| \) reflects values symmetrically around \( x = -2 \). It is piecewise and has a "V" shape.
   - The negative sign in front of the absolute value \( -|x + 2| \) inverts the "V" shape, turning it into a downward-facing "V".
   - Finally, subtracting 2 from the entire expression \( -|x + 2| - 2 \) shifts the graph 2 units downward.

### 2. **Vertex of the function**:
   - The vertex of the function occurs where \( |x + 2| = 0 \), i.e., when \( x = -2 \).
   - Substituting \( x = -2 \) into the function, we get:
     \[
     f(-2) = -| -2 + 2 | - 2 = -0 - 2 = -2
     \]
   The vertex is at \( (-2, -2) \).

### 3. **Behavior of the function**:
   - As \( x \) moves away from \( -2 \), the value of \( |x + 2| \) increases. Since the absolute value is multiplied by -1, the function decreases.
   - The function has no upper bound, since it is always decreasing as we move away from \( x = -2 \).

### 4. **Range of the function**:
   - The maximum value of \( f(x) \) occurs at the vertex, which is \( -2 \).
   - The function decreases without bound as \( x \) moves away from the vertex.
   - Therefore, the range of the function is:
     \[
     (-\infty, -2]
     \]

If you were to graph this function using a graphing tool, you'd observe a downward-facing "V" with a maximum at \( (-2, -2) \) and the rest of the graph extending downward.

Would you like more details or have any questions?

### Related Questions:
1. How would the graph change if the absolute value term were \( |x - 2| \) instead of \( |x + 2| \)?
2. What is the domain of the function \( f(x) = -|x + 2| - 2 \)?
3. How would the range change if we added a constant \( c \) to the function \( f(x) = -|x + 2| - 2 + c \)?
4. Can you find the derivative of \( f(x) = -|x + 2| - 2 \), and what does it tell us about the function's behavior?
5. How would the function behave if the absolute value term were multiplied by a positive constant (e.g., \( -2|x + 2| - 2 \))?

### Tip:
Always check the vertex of an absolute value function when looking for the maximum or minimum values; it helps in determining the range.

Use graphing technology to find the range of the function f(x) = -|x + 2| - 2.

Discover how to use graphing tools to find the range of the function f(x) = -|x + 2| - 2. This guide covers algebraic concepts, properties of absolute value functions, and the graphical approach to determining the range. Suitable for students in Grades 9-11.

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