https://file.aiquickdraw.com/mathbot/file/17282638909281hg45lws.jpg

Let's analyze the relationship between the graphs of \( f(x) \) and \( g(x) \) using the provided information.

1. **Graph transformation:** Looking at the two graphs, it seems that \( g(x) \) is a reflection of \( f(x) \) over the y-axis. This reflection is a horizontal flip. The main indicator is that the overall shape of \( g(x) \) is similar to \( f(x) \), but it has been mirrored over the y-axis.

2. **Points comparison:** The graphs indicate that:
   - \( f(x) \) ends at \( (4, 6) \), while \( g(x) \) ends at \( (2, 6) \). This suggests a horizontal compression. Specifically, the x-coordinates have been scaled by a factor, possibly indicating a horizontal transformation.

3. **Reflection and scaling:** Since we notice that the x-values are halved when going from \( f(x) \) to \( g(x) \), the x-coordinates have likely been multiplied by \( \frac{1}{2} \).

4. **Effect on points:** We should also check specific values. For instance, if \( f(-2) \) corresponds to some value, we can see if \( g(-1) \) shares the same value, which would confirm the horizontal compression.

Thus, based on the given graphs:

- The graph of \( g(x) \) is **a horizontal reflection** of the graph of \( f(x) \).
- The x-values have been **halved**.
- This is the result of a horizontal compression by a factor of **2**.

Would you like more detailed steps on how these transformations are applied to functions?

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Here are five related questions you might find useful:
1. How can we mathematically express the horizontal reflection and compression in a transformation equation?
2. What other types of transformations (e.g., vertical shifts) might apply to similar problems?
3. If \( f(x) \) is shifted vertically, how would \( g(x) \) be affected by that?
4. How would the derivative of \( f(x) \) change under a horizontal compression?
5. Can we generalize this transformation for all types of functions, or is it specific to the given case?

**Tip:** When identifying transformations between two graphs, always compare key points (like intercepts and endpoints) to understand scaling or reflections.

Two graphs are shown below. The f(x) graph is the original graph. Complete the statements below to describe the relationship between the functions f(x) and g(x).

Explore the relationship between the graphs of f(x) and g(x) through transformations like horizontal reflections and compressions. This detailed analysis focuses on identifying key changes in function graphs, suitable for students in Grades 9-11.

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