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The question asks about the transformation of the graph of \( y = f(x) \), shown as a V-shaped curve with its vertex at the origin, and the resulting graph of \( y = f(x + 1) \). This involves a **horizontal translation** of the original graph.

### Transformation Analysis:
The equation \( y = f(x + 1) \) represents a **leftward shift** of the graph by 1 unit. In general, for any function \( f(x) \), replacing \( x \) with \( x + 1 \) moves the graph 1 unit to the left.

### Expected Transformation:
- The original graph of \( y = f(x) \) has a vertex at \( (0, 0) \).
- After shifting left by 1 unit, the new vertex of the transformed graph will be at \( (-1, 0) \).

### Answer Choice:
Looking at the options, you should select the graph that corresponds to the V-shaped curve with the vertex at \( (-1, 0) \), which matches option (D).

Thus, **option (D)** shows the correct graph of \( y = f(x + 1) \).

Would you like more details or have any questions? Here are five related questions to further explore this topic:

1. How does a vertical translation affect a function?
2. What happens when you apply a horizontal translation to a quadratic function?
3. How can you tell whether a graph has been reflected across the x-axis or y-axis?
4. How does the transformation \( y = f(x - h) \) affect a graph?
5. Can you describe the effect of both vertical and horizontal shifts on a given graph?

**Tip:** Always remember, adding a constant inside the function (to \( x \)) results in a horizontal shift, while adding it outside results in a vertical shift!

The graph of y = f(x) is shown above. Which of the following could be the graph of y = f(x + 1)?

Explore how to shift a graph of the function y = f(x) horizontally by modifying the function to y = f(x + 1). Learn how this transformation affects the graph's position with detailed examples. Suitable for high school students in Grades 9-12.

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