To identify which quadratic function represents the given parabola, let's analyze its properties based on the graph:

1. Vertex:
   The vertex of the parabola is at $(x, y) = (-1, 6)$. This suggests that the equation of the parabola is in the vertex form:

   $f(x) = a(x - h)^2 + k$

   where $(h, k)$ is the vertex. For this graph, $h = -1$ and $k = 6$.

2. Direction and Coefficient:
   The parabola opens downwards, indicating a negative leading coefficient $a$. Also, the parabola appears to be vertically compressed, suggesting that $a < 0$ and not equal to -1.

3. Matching with Options:

   • The correct equation should have the form:

     $f(x) = -a(x + 1)^2 + 6$

   where $a$ is negative. Among the options, the function that matches these criteria is:

   $f(x) = -0.5(x + 1)^2 + 6$.

Final Answer:

The quadratic function that represents the given parabola is:

$f(x) = -0.5(x + 1)^2 + 6$.

Would you like further details or have any questions?

Here are 5 follow-up questions to consider:

1. What is the significance of the vertex in the vertex form of a quadratic function?
2. How does changing the coefficient $a$ affect the shape of a parabola?
3. What other properties can be used to identify a quadratic function from its graph?
4. How would you convert a quadratic function from vertex form to standard form?
5. What transformations occur if you change the values of $h$ and $k$ in the vertex form?

Tip: When working with quadratic functions in vertex form, remember that the sign of $a$ determines the direction of the parabola, while the values of $h$ and $k$ specify its position on the graph.