The graph shown is a parabola, and we are tasked with finding the quadratic function $f(x)$. From the graph, we can observe the following key points:

1. The vertex of the parabola is at $(-1, -4)$, which gives us important information for writing the function in vertex form: $f(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex. In this case, $h = -1$ and $k = -4$, so the equation becomes: $f(x) = a(x + 1)^2 - 4$

2. Next, we use the additional point $(-2, -2)$ provided in the graph to solve for $a$. Substitute $x = -2$ and $f(x) = -2$ into the equation: $-2 = a(-2 + 1)^2 - 4$ Simplifying: $-2 = a(1)^2 - 4$ $-2 = a - 4$ Solving for $a$: $a = 2$

3. Therefore, the equation of the quadratic function is: $f(x) = 2(x + 1)^2 - 4$

Would you like more details or have any further questions?

Here are 5 related questions to expand this topic:

1. What is the standard form of the quadratic function from the vertex form we found?
2. How do you find the axis of symmetry from the equation of a quadratic function?
3. What is the importance of the discriminant in quadratic functions?
4. Can a parabola be translated without changing its vertex?
5. How does the value of $a$ affect the direction and width of the parabola?

Tip: Remember that the sign of $a$ in a quadratic equation determines whether the parabola opens upward or downward!