From the graph provided, we have two parabolas. The solid black graph represents the function $y = f(x)$, and the dotted graph is a transformation of this function. Let's analyze the transformation:

1. Horizontal Translation: The dotted graph is shifted to the left compared to the solid graph, which indicates a horizontal translation.

2. Vertical Translation: The dotted graph is also shifted downward, suggesting a vertical translation.

The transformation rules are:

• Horizontal shifts: If the graph shifts left by $a$, it is represented by $f(x + a)$.
• Vertical shifts: If the graph shifts down by $b$, it is represented by $f(x) - b$.

By observing the graph:

• The dotted graph has been shifted 4 units to the left.
• The dotted graph has also been shifted 4 units down.

Therefore, the correct function representing the dotted graph is: $y = f(x + 4) - 4$

This corresponds to the third option: $y = f(x+4) - 4$.

Would you like further explanation on function transformations?

Here are five follow-up questions related to this concept:

1. What happens if you shift the graph $f(x)$ to the right instead of the left?
2. How does a vertical stretch or compression affect the graph of $f(x)$?
3. How would the equation change if the graph were shifted upwards instead of downwards?
4. Can you describe how a reflection over the x-axis would affect $f(x)$?
5. How would both a horizontal stretch and vertical stretch affect the graph of $f(x)$?

Tip:

To quickly identify horizontal shifts, remember that moving left corresponds to $f(x + a)$, while moving right corresponds to $f(x - a)$.