Let's go through each part of the question based on the graph provided:

(a) Find the domain of $f$:

• The graph shows that the function $f(x)$ exists for all values of $x$ from $x = -2$ to $x = 12$.
• Therefore, the domain of $f$ is $[-2, 12]$.

(b) Find the range of $f$:

• The minimum value of $f(x)$ on the graph is at $y = -4$ and the maximum value is at $y = 9$.
• Therefore, the range of $f$ is $[-4, 9]$.

(c) Find $f(-2)$:

• From the graph, at $x = -2$, the function $f(x)$ is at $y = 9$.
• So, $f(-2) = 9$.

(d) Find the values of $x$ for which $f(x) = -6$:

• Looking at the graph, $f(x)$ never reaches $-6$. The lowest point of the graph is $y = -4$.
• Therefore, there are no values of $x$ for which $f(x) = -6$.

(e) Find the points where the graph of $f$ crosses the $x$-axis:

• The graph crosses the $x$-axis at $x = 2$ and $x = 8$.
• Therefore, the points are $(2, 0)$ and $(8, 0)$.

Would you like further details or have any questions?

Here are some related questions for deeper understanding:

1. What is the vertex form of a parabola, and can it be determined from this graph?
2. How do you find the axis of symmetry of a parabolic function like this one?
3. What are the steps to determine the domain and range of more complex functions?
4. How can we algebraically verify the points where a function crosses the $x$-axis?
5. What happens to the domain and range if this graph was shifted vertically or horizontally?

Tip: When analyzing a graph, always note the critical points such as intercepts, maximums, minimums, and endpoints to understand the function's behavior fully.