Let's analyze the given function step by step.
The function provided is [tex]\( f(x) = -(x + 2)^2 - 1 \)[/tex].
1. Rewrite the function in vertex form:
The given function is already in vertex form:
[tex]\[
y = a(x - h)^2 + k
\][/tex]
Here, [tex]\( a = -1 \)[/tex], [tex]\( h = -2 \)[/tex], and [tex]\( k = -1 \)[/tex]. Therefore, the function is:
[tex]\[
y = -(x + 2)^2 - 1
\][/tex]
2. Determine the vertex:
The vertex of a parabola in the form [tex]\( y = a(x - h)^2 + k \)[/tex] is given by the point [tex]\( (h, k) \)[/tex].
Here, [tex]\( h = -2 \)[/tex] and [tex]\( k = -1 \)[/tex]. Therefore, the vertex of the function is:
[tex]\[
(-2, -1)
\][/tex]
3. Find the axis of symmetry:
The axis of symmetry for a parabola in the form [tex]\( y = a(x - h)^2 + k \)[/tex] is the vertical line given by [tex]\( x = h \)[/tex].
Here, [tex]\( h = -2 \)[/tex]. Therefore, the equation for the axis of symmetry is:
[tex]\[
x = -2
\][/tex]
4. Determine if the vertex is a maximum or minimum:
The coefficient [tex]\( a \)[/tex] determines whether the parabola opens upwards or downwards.
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards, and the vertex is a minimum.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards, and the vertex is a maximum.
Here, [tex]\( a = -1 \)[/tex], which is less than 0, indicating that the parabola opens downwards. Therefore, the vertex at [tex]\( (-2, -1) \)[/tex] is a maximum.
Next, let us analyze the provided options considering our conclusions about the function:
1. The axis of symmetry should be [tex]\( x = -1 \)[/tex]: This is incorrect. The axis of symmetry is [tex]\( x = -2 \)[/tex].
2. The axis of symmetry should be [tex]\( x = 2 \)[/tex]: This is also incorrect. As stated earlier, the axis of symmetry is [tex]\( x = -2 \)[/tex].
3. The vertex should be a maximum: This is correct. Since the coefficient [tex]\( a \)[/tex] is negative, the parabola opens downward, making the vertex a maximum.
4. The vertex should be [tex]\((-2, 1)\)[/tex]: This is incorrect. The correct vertex is [tex]\((-2, -1)\)[/tex].
Therefore, the best description of the error in Ali's graph is:
The vertex should be a maximum.
Thus, the correct option is:
The vertex should be a maximum.
