https://file.aiquickdraw.com/mathbot/file/1728332571472a0eiup9s.jpg

To analyze the graph in the image, let's proceed step-by-step:

1. The solid black graph represents the original function \( y = f(x) \), which seems to be a straight line passing through the origin with a positive slope.
2. The dotted graph appears to be a reflection and transformation of the solid graph.

### Understanding the Transformation
From the given options, let's analyze how each transformation would affect the graph:

- **Option A: \( y = -f(x + 4) \)**
  - The \( x + 4 \) indicates a horizontal shift to the left by 4 units.
  - The negative sign in front of \( f \) means the graph is reflected over the x-axis.

- **Option B: \( y = -f(x) - 4 \)**
  - The negative sign means reflection over the x-axis.
  - The \( -4 \) outside means a vertical shift downward by 4 units.

- **Option C: \( y = -f(x) + 4 \)**
  - The negative sign means reflection over the x-axis.
  - The \( +4 \) means a vertical shift upward by 4 units.

- **Option D: \( y = -f(x - 4) \)**
  - The \( x - 4 \) indicates a horizontal shift to the right by 4 units.
  - The negative sign means reflection over the x-axis.

### Observing the Dotted Graph
The dotted graph is a reflection of the original function \( f(x) \) across the x-axis and is shifted to the right by 4 units. Based on this observation, the correct transformation is represented by:

**Option D: \( y = -f(x - 4) \)**

This is the answer that matches the transformation described in the graph.

Would you like more details on how these transformations work, or do you have any questions?

Here are 5 questions you might be interested in:
1. What is the general effect of reflecting a function over the x-axis?
2. How do horizontal shifts affect the graph of a function?
3. What are vertical shifts, and how do they differ from horizontal shifts in graph transformations?
4. Can you explain how to determine the equation of a transformed graph given its original form?
5. How do combined transformations affect the order in which you apply them?

**Tip:** Always perform horizontal transformations before vertical ones when analyzing the effects on the graph.

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

Learn how to analyze the transformation of a graph, including reflection over the x-axis and horizontal shifts. This example involves transforming the graph of y = f(x) to y = -f(x - 4). Suitable for high school students.

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