Answer :
To determine the interval over which the graph of the function \( f(x) = -(x + 8)^2 - 1 \) is decreasing, we need to analyze the properties and behavior of this function. Here is a step-by-step explanation:
1. Determine the form of the function: The given function is \( f(x) = -(x + 8)^2 - 1 \). This function represents a parabola.
2. Identify the direction of the parabola: Since the coefficient of the \((x + 8)^2\) term is negative (i.e., \(-(x + 8)^2\)), this is a downward-opening parabola.
3. Find the vertex of the parabola:
- The vertex form of a parabola is given by \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
- For the given function, \( f(x) = -(x + 8)^2 - 1 \), we can rewrite it as \( f(x) = -(x - (-8))^2 - 1 \).
- Therefore, the vertex \((h, k)\) is at \((-8, -1)\).
4. Behavior of the parabola around the vertex:
- For a downward-opening parabola, it decreases to the left of the vertex and increases to the right of the vertex.
5. Identify the decreasing interval:
- Since the vertex is at \( x = -8 \) and the parabola opens downward, the function decreases for all \( x \) values to the left of \(-8\).
- Hence, the interval over which the graph of the function is decreasing is \( (-\infty, -8) \).
So, the correct interval over which the graph of \( f(x) = -(x + 8)^2 - 1 \) is decreasing is:
[tex]\[ (-\infty, -8) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{(-\infty, -8)} \][/tex]
1. Determine the form of the function: The given function is \( f(x) = -(x + 8)^2 - 1 \). This function represents a parabola.
2. Identify the direction of the parabola: Since the coefficient of the \((x + 8)^2\) term is negative (i.e., \(-(x + 8)^2\)), this is a downward-opening parabola.
3. Find the vertex of the parabola:
- The vertex form of a parabola is given by \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
- For the given function, \( f(x) = -(x + 8)^2 - 1 \), we can rewrite it as \( f(x) = -(x - (-8))^2 - 1 \).
- Therefore, the vertex \((h, k)\) is at \((-8, -1)\).
4. Behavior of the parabola around the vertex:
- For a downward-opening parabola, it decreases to the left of the vertex and increases to the right of the vertex.
5. Identify the decreasing interval:
- Since the vertex is at \( x = -8 \) and the parabola opens downward, the function decreases for all \( x \) values to the left of \(-8\).
- Hence, the interval over which the graph of the function is decreasing is \( (-\infty, -8) \).
So, the correct interval over which the graph of \( f(x) = -(x + 8)^2 - 1 \) is decreasing is:
[tex]\[ (-\infty, -8) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{(-\infty, -8)} \][/tex]
