https://file.aiquickdraw.com/mathbot/file/1725447280983xanhflg2.jpg

The image is a problem about function transformations. The graph of \(y = f(x)\) is given, and the task is to draw the graph of \(y = -\frac{1}{2}f(x)\).

To solve this problem, we need to understand how the given transformation affects the original graph:

1. **Reflection across the x-axis:** The negative sign in \(-\frac{1}{2}f(x)\) indicates a reflection over the x-axis. This means that every point on the graph of \(f(x)\) will be mirrored across the x-axis.
2. **Vertical compression by a factor of \(\frac{1}{2}\):** The factor \(\frac{1}{2}\) indicates a vertical compression. This means that the y-coordinates of all points on the reflected graph will be halved.

### Steps to Draw the Graph:
1. **Identify the key points** on the original graph \(y = f(x)\). Let's assume we have points like \((x_1, y_1)\), \((x_2, y_2)\), etc.
2. **Apply the reflection** across the x-axis: Each point \((x_i, y_i)\) will become \((x_i, -y_i)\).
3. **Apply the vertical compression**: Multiply the y-coordinates by \(\frac{1}{2}\). The reflected points will transform from \((x_i, -y_i)\) to \((x_i, -\frac{1}{2}y_i)\).

### Applying the Transformation to Each Point:
- Take each point from the graph of \(y = f(x)\) and apply the transformations described above.
- Plot these new points on the coordinate plane to draw the graph of \(y = -\frac{1}{2}f(x)\).

Would you like a more detailed explanation or have any questions about this transformation process?

Here are some related questions to explore:
1. What happens if the transformation was \(y = \frac{1}{2}f(x)\) instead?
2. How does a horizontal stretch or compression affect the graph?
3. What is the effect of a translation of the function, such as \(y = f(x) + c\)?
4. How would the transformation change if the expression was \(y = -2f(x)\)?
5. How can we determine the new range of the function after applying transformations?

**Tip:** Understanding each transformation (reflection, vertical/horizontal stretch, and translation) separately can help in combining them correctly to draw the transformed graph accurately.

Below is the graph of y = f(x). Using the red points to draw the graph of y = -\frac{1}{2}f(x).

Understand how to transform the graph of y = f(x) to y = -\frac{1}{2}f(x) by applying reflection across the x-axis and vertical compression. This problem is suitable for students in Grades 9-11.

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