Question

sanjay begins to correctly graph the function ( f(x) = (x + 1)^2 - 3 ). based on the axis of symmetry and the vertex, which graph could be sanjay’s?

Explanation:

Step1: Recall the vertex form of a parabola

The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola, and the axis of symmetry is \( x = h \).

Step2: Identify the vertex and axis of symmetry for \( f(x) = (x + 1)^2 - 3 \)

For the function \( f(x) = (x + 1)^2 - 3 \), we can rewrite it as \( f(x) = (x - (-1))^2 + (-3) \). So, the vertex \((h, k)\) is \((-1, -3)\), and the axis of symmetry is \( x = -1 \).

Step3: Analyze the graphs

We need to find the graph where the vertex is at \((-1, -3)\) and the axis of symmetry is \( x = -1 \). Looking at the options, we check the coordinates of the vertex and the axis of symmetry. The graph with vertex \((-1, -3)\) and axis of symmetry \( x = -1 \) (the one with the vertex at \((-1, -3)\) and the dashed line at \( x = -1 \)) is the correct one. (Assuming from the description, the second graph or the one with vertex \((-1, -3)\) is the answer, but based on the vertex form analysis, the vertex is \((-1, -3)\) and axis \( x=-1 \), so we identify the graph with these properties.)

Answer:

The graph with vertex \((-1, -3)\) and axis of symmetry \( x = -1 \) (the one where the vertex is marked as \((-1, -3)\) and the dashed line is \( x = -1 \)) is Sanjay's graph. (If we consider the options, the one with vertex \((-1, -3)\) is the correct graph, so depending on the labels, for example, if the second graph has vertex \((-1, -3)\), then that's the answer. But based on the function, the vertex is \((-1, -3)\) and axis \( x=-1 \), so the graph with these features is the correct one.)