The graph shows a V-shaped curve, which suggests it is an absolute value function of the form $f(x) = a|x - h| + k$, where $(h, k)$ is the vertex of the function, and $a$ affects the slope and direction of the lines (whether the V opens upwards or downwards).

Key observations:

• The vertex is at $(3, -3)$.
• The graph opens downwards, meaning $a$ must be negative.
• The function is shifted 3 units to the right (so the expression should be $|x - 3|$).

Analyzing the given options:

1. $f(x) = -3|x - 3| - 3$ seems like a good fit, as it reflects the downward opening, the vertex at $(3, -3)$, and the correct transformation.
2. The other options either have incorrect signs for $a$ (like the positive ones that would open upwards), or incorrect horizontal shifts (like those involving $x + 3$).

Thus, the correct answer is:

$f(x) = -3|x - 3| - 3$

Would you like further details or clarifications?

Related Questions:

1. How do you find the vertex of an absolute value function?
2. How does the coefficient $a$ in $f(x) = a|x - h| + k$ affect the graph?
3. What is the effect of the transformation $|x - h|$ on the graph of the absolute value function?
4. How can you determine whether the graph opens upwards or downwards?
5. How do shifts in $x$ and $y$ change the position of the graph?

Tip:

When solving absolute value problems, always identify the vertex and check the direction (upwards or downwards) to quickly narrow down the correct function.