Q. Ali graphs the function ( f(x) = -(x + 2)^2 – 1 ) as shown. Which best describes the error in the graph?

Write the function and identify the vertex form. A parabola in vertex form is \(f(x)=a(x-h)^{2}+k\), where the vertex is \((h,k)\) and the axis of symmetry is \(x=h\). The given function is

\(f(x)=-(x+2)^{2}-1\).

Match the given function to the vertex form by recognizing \(x+2=x-(-2)\). Thus

\(f(x)=-1\bigl(x-(-2)\bigr)^{2}+(-1)\).

So \(a=-1\), \(h=-2\), and \(k=-1\). Therefore the vertex is \((-2,-1)\).

Determine the axis of symmetry. The axis is \(x=h\), so the axis of symmetry is

\(x=-2\).

Determine whether the vertex is a maximum or minimum. The sign of \(a\) tells the opening direction: if \(a>0\) the parabola opens upward (vertex is a minimum); if \(a<0\) it opens downward (vertex is a maximum). Here \(a=-1<0\), so the parabola opens downward and the vertex is a maximum.

Compare with the answer choices:

• Axis should be \(x=-1\): incorrect (actual axis is \(x=-2\)).
• Axis should be \(x=2\): incorrect.
• Vertex should be a maximum: correct, because \(a=-1<0\).
• Vertex should be \((-2,1)\): incorrect (the correct vertex is \((-2,-1)\)).

Conclusion: the best description of the error in the graph is that the vertex should be a maximum.