https://file.aiquickdraw.com/mathbot/file/1729152580795l4675ydy.jpg

Let's analyze each function transformation step by step:

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### a. \( y = f(x) - 1 \)
- **Transformation**: This is a **vertical shift downward** by 1 unit.
  - Every point on the graph of \( f(x) \) is shifted down by 1 unit.
  
**Result**: The new graph moves vertically down.

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### b. \( y = \frac{1}{2}(f(x) + 2) \)
- First: \( f(x) + 2 \) indicates a **vertical shift upward** by 2 units.
  - This moves the graph of \( f(x) \) 2 units **up**.
- Then: \( \frac{1}{2} \) scales the graph **vertically by a factor of 1/2** (compression).
  - This brings all the points closer to the x-axis.

**Result**: The graph shifts 2 units up and is vertically compressed.

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### c. \( y = -2(f(x) - 5) \)
- First: \( f(x) - 5 \) shifts the graph **downward by 5 units**.
- Then: \( -2 \) applies two transformations:
  1. The negative sign reflects the graph across the x-axis.
  2. The factor 2 **stretches the graph vertically** by a factor of 2.

**Result**: The graph is shifted down by 5 units, reflected across the x-axis, and stretched vertically.

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Would you like further clarification or a detailed graph for any of these transformations?

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### Related Questions:
1. What is the difference between vertical stretching and compression?
2. How does adding a constant inside vs. outside the function change the transformations?
3. What happens if the reflection is applied without any stretching?
4. How would the transformations change if the coefficient in part (b) were negative?
5. How can horizontal shifts and reflections be applied to the function?

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**Tip**: A general rule is that constants **outside** the function affect vertical changes, while constants **inside** affect horizontal shifts.

In each of the following functions, describe how the graph of the given function can be obtained from the graph of f(x). a. y = f(x) - 1 b. y = (1/2)(f(x) + 2) c. y = -2(f(x) - 5)

Learn how to perform graph transformations on functions. This problem covers vertical shifts, compression, and reflections, with step-by-step explanations on how each transformation affects the graph of f(x). Suitable for Grades 9-12.

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