The problem asks to identify which function represents the transformation of the graph $y = f(x)$, where the solid black graph is the original, and the dotted graph is the transformed one.

Observing the transformations:

1. Horizontal Shift:

   • The dotted graph seems to have been shifted left compared to the solid graph. This implies a horizontal shift. For left shifts, the function becomes $y = f(x + h)$, where $h$ is positive.

2. Vertical Shift:

   • The dotted graph is also shifted upwards compared to the original graph. An upward shift corresponds to adding a constant $k$ to the function, i.e., $y = f(x) + k$, where $k > 0$.

Determining the specific transformations:

• Horizontal shift: It looks like the graph has shifted 3 units to the left.
• Vertical shift: The graph has been shifted 4 units upwards.

Thus, the function that represents the dotted graph should be:

$y = f(x + 3) + 4$

Answer:

The correct choice is $y = f(x + 3) + 4$.

Do you want more details on transformations or have any other questions?

Here are five related questions for further exploration:

1. How does shifting a graph horizontally affect the function's equation?
2. What is the difference between horizontal and vertical shifts?
3. How do reflections over the x-axis or y-axis change the function's graph?
4. Can you describe how to identify a vertical stretch or compression from a graph?
5. How do combinations of transformations (e.g., shifts and stretches) interact on a graph?

Tip: When working with graph transformations, always observe the direction of shifts and the size of the shift from reference points on the original graph.