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.