https://file.aiquickdraw.com/mathbot/file/1728577449578a3an8bhs.jpg

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 **\( y = f(x + 3) + 4 \)**.

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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.

The graph of y = f(x) is the solid black graph below. Which function represents the dotted graph?

Learn how to identify a transformed function from a shifted graph. This problem involves analyzing horizontal and vertical shifts in function graphs, using algebraic concepts. Suitable for students in Grades 9-11.

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