To analyze the given function [tex]\( f(x) = -(x+2)^2 - 1 \)[/tex] and determine the error in the graph, let's break it down step by step.
1. Identifying the Form of the Function:
The function [tex]\( f(x) = -(x+2)^2 - 1 \)[/tex] is in vertex form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\(a\)[/tex] determines the direction the parabola opens.
Here, [tex]\(a = -1\)[/tex], [tex]\(h = -2\)[/tex], and [tex]\(k = -1\)[/tex].
2. Determining the Vertex:
The vertex of the function [tex]\( f(x) = -(x+2)^2 - 1 \)[/tex] is [tex]\((-2, -1)\)[/tex].
3. Axis of Symmetry:
The axis of symmetry for a parabola in the form [tex]\( f(x) = a(x-h)^2 + k \)[/tex] is the vertical line [tex]\( x = h \)[/tex].
Since [tex]\( h = -2 \)[/tex], the axis of symmetry is [tex]\( x = -2 \)[/tex].
4. Nature of the Vertex (Maximum or Minimum):
- The value of [tex]\( a = -1 \)[/tex] is negative, indicating that the parabola opens downwards.
- Therefore, the vertex represents a maximum point.
With this information, we can evaluate the options presented:
1. The axis of symmetry should be [tex]\( x = -1 \)[/tex]:
- This is incorrect based on our calculation, where the correct axis of symmetry is [tex]\( x = -2 \)[/tex].
2. The axis of symmetry should be [tex]\( x = 2 \)[/tex]:
- This is also incorrect. The correct axis of symmetry is [tex]\( x = -2 \)[/tex].
3. The vertex should be a maximum:
- This is correct. Because the parabola opens downwards, the vertex at [tex]\((-2, -1)\)[/tex] is indeed a maximum.
4. The vertex should be [tex]\((-2, 1)\)[/tex]:
- This is incorrect. The correct vertex is [tex]\((-2, -1)\)[/tex].
After analyzing all given options, it is evident that the correct statement about the graph's error is:
The vertex should be a maximum.
