Let's analyze the function [tex]\( f(x) = -2(x + 3)^2 - 1 \)[/tex] step-by-step to identify the vertex, domain, and range.
1. Vertex:
- The given function is in the form [tex]\( f(x) = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex.
- In our equation [tex]\( f(x) = -2(x + 3)^2 - 1 \)[/tex], we can see that:
[tex]\[ a = -2 \][/tex]
[tex]\[ h = -3 \][/tex] (note the sign change inside the parentheses; [tex]\( (x + 3) \)[/tex] means [tex]\( h = -3 \)[/tex])
[tex]\[ k = -1 \][/tex]
- Therefore, the vertex of the function is [tex]\((-3, -1)\)[/tex].
2. Domain:
- The domain of any quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] is all real numbers, because you can input any real number [tex]\( x \)[/tex] and get a corresponding [tex]\( f(x) \)[/tex].
- Thus, the domain of [tex]\( f(x) = -2(x + 3)^2 - 1 \)[/tex] is all real numbers.
3. Range:
- Since the coefficient of the squared term [tex]\(( -2 )\)[/tex] is negative, the parabola opens downwards.
- The vertex [tex]\((-3, -1)\)[/tex] is the maximum point on the graph of the function, since the parabola opens downwards.
- Therefore, the range of the function includes all [tex]\( y \)[/tex]-values that are less than or equal to the vertex’s [tex]\( y \)[/tex]-coordinate [tex]\((-1)\)[/tex].
- Hence, the range is [tex]\( y \leq -1 \)[/tex].
Combining these results, we have:
- The vertex is [tex]\((-3, -1)\)[/tex].
- The domain is all real numbers.
- The range is [tex]\( y \leq -1 \)[/tex].
So, the correct choice is:
The vertex is [tex]\((-3, -1)\)[/tex], the domain is all real numbers, and the range is [tex]\( y \leq -1 \)[/tex].
