Question

phs the function $f(x) = -(x + 2)^2 - 1$ as shown. which best describes the error in the graph? the axis of symmetry should be $x = 2$. the vertex should be a maximum. the vertex should be $(-2, 1)$. the axis of symmetry should be $x = -1$.

Explanation:

1. Recall the vertex form of a parabola: \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and the axis of symmetry is \( x = h \). The coefficient \( a \) determines if the parabola opens up ( \( a>0 \), minimum vertex) or down ( \( a<0 \), maximum vertex).
2. For the function \( f(x) = -(x + 2)^2 - 1 \), we can rewrite it as \( f(x) = -1(x - (-2))^2 + (-1) \). So, \( a=-1 \), \( h = -2 \), \( k=-1 \).
3. Analyze each option:

• Option 1: The axis of symmetry should be \( x = h=-2 \), not \( x = 2 \). So this is incorrect.
• Option 2: Since \( a=-1<0 \), the parabola opens downward, so the vertex is a maximum. In the graph, the parabola opens upward (which is wrong), so the vertex should be a maximum. This is correct.
• Option 3: The vertex is \((h,k)=(-2,-1)\), not \((-2,1)\). So this is incorrect.
• Option 4: The axis of symmetry is \( x = h=-2 \), not \( x=-1 \). So this is incorrect.

Answer:

The vertex should be a maximum.