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The problem in question asks for several things regarding the function:

- **Find the minimum or maximum value of the function.**
- **Describe the domain and range of the function.**
- **Determine where the function is increasing and decreasing.**

Let’s go over problem **41**, where the function is:

\[
y = -x^2 - 4x - 2
\]

### 1. Find the Minimum or Maximum Value of the Function
This is a quadratic function in the form \(y = ax^2 + bx + c\), where \(a = -1\), \(b = -4\), and \(c = -2\).

Since \(a = -1\) (negative), the parabola opens **downward**, meaning the function has a **maximum value**.

To find the maximum value, use the vertex formula for the x-coordinate of the vertex:
\[
x = \frac{-b}{2a} = \frac{-(-4)}{2(-1)} = \frac{4}{-2} = -2
\]

Now, substitute \(x = -2\) into the function to find the y-coordinate of the vertex (the maximum value):
\[
y = -(-2)^2 - 4(-2) - 2 = -4 + 8 - 2 = 2
\]
So, the maximum value is **2**.

### 2. Describe the Domain and Range
- The **domain** of any quadratic function is all real numbers: \(\mathbb{R}\), or \((- \infty, \infty)\).
- The **range** depends on whether the parabola opens up or down. Since the parabola opens downward and has a maximum value of 2, the range is:
  \[
  (-\infty, 2]
  \]

### 3. Determine Where the Function is Increasing and Decreasing
For a parabola opening downward:
- The function is **increasing** on the interval to the left of the vertex, \( (-\infty, -2) \).
- The function is **decreasing** on the interval to the right of the vertex, \( (-2, \infty) \).

### Summary of Key Points:
- **Maximum value**: 2 (occurs at \(x = -2\)).
- **Domain**: \((- \infty, \infty)\)
- **Range**: \((- \infty, 2]\)
- **Increasing**: \( (-\infty, -2) \)
- **Decreasing**: \( (-2, \infty) \)

Would you like more details on any specific part of this solution?  
Here are some related questions for further practice:

1. How do you find the vertex of a quadratic function?
2. What is the significance of the sign of the leading coefficient in determining the direction of a parabola?
3. How do you determine if a quadratic function has a minimum or maximum value?
4. Can you explain the relationship between the vertex and the intervals of increase/decrease for a parabola?
5. How do we define the domain and range for any quadratic function?

**Tip**: The vertex of a parabola provides both the maximum or minimum value and splits the graph into increasing and decreasing intervals.

Find the minimum or maximum value of the function y = -x^2 - 4x - 2. Describe the domain and range of the function, and where the function is increasing and decreasing.

Solve the quadratic function y = -x^2 - 4x - 2, find its maximum value, describe the domain and range, and determine where the function is increasing and decreasing. This solution includes the use of the vertex formula and explores the behavior of the parabola. Suitable for students in Grades 9-11.

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