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For **Problem 13**, you're given a graph of a function \( f(x) \) with specific points and asked to sketch the transformations of this function based on different formulas. I'll break down each transformation:

1. **(a) \( y = f(x) + 2 \)**: This is a vertical shift upwards by 2 units. All points on the graph move 2 units up.
   
2. **(b) \( y = -f(x) \)**: This is a reflection of the graph across the x-axis. All the y-values of the points change sign.
   
3. **(c) \( y = f(x - 2) \)**: This is a horizontal shift to the right by 2 units. All points on the graph move 2 units to the right.

4. **(d) \( y = f(x + 3) \)**: This is a horizontal shift to the left by 3 units. All points on the graph move 3 units to the left.

5. **(e) \( y = 2f(x) \)**: This is a vertical stretch by a factor of 2. The distance of each point from the x-axis doubles.

6. **(f) \( y = f(-x) \)**: This is a reflection of the graph across the y-axis. The x-values change sign, reflecting the graph horizontally.

7. **(g) \( y = f\left(\frac{1}{2}x\right) \)**: This is a horizontal stretch by a factor of 2. The graph stretches out, making it wider.

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For **Problem 14**, the function \( f(x) \) is given with different transformation options:

1. **(a) \( y = f(x) - 1 \)**: This is a vertical shift downwards by 1 unit. All points on the graph move 1 unit down.

2. **(b) \( y = f(x + 1) \)**: This is a horizontal shift to the left by 1 unit. All points on the graph move 1 unit to the left.

3. **(c) \( y = f(x - 1) \)**: This is a horizontal shift to the right by 1 unit. All points move 1 unit to the right.

4. **(d) \( y = -f(x - 2) \)**: This combines a horizontal shift to the right by 2 units and a reflection across the x-axis.

5. **(e) \( y = f(-x) \)**: This is a reflection of the graph across the y-axis. The x-values change sign.

6. **(f) \( y = \frac{1}{2}f(x) \)**: This is a vertical compression by a factor of \(\frac{1}{2}\), meaning the y-values are halved, bringing the graph closer to the x-axis.

7. **(g) \( y = f(2x) \)**: This is a horizontal compression by a factor of 2, making the graph narrower.

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### Do you want further details or have any questions on how these transformations affect the graph?

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Here are 5 related questions to explore:

1. How does a horizontal shift differ from a vertical shift in terms of affecting the graph?
2. What would happen if you applied both a horizontal and vertical shift to the function?
3. How do reflections across the x-axis and y-axis affect the graph's symmetry?
4. What is the effect of combining multiple transformations (e.g., stretch and shift)?
5. How does a horizontal stretch/compression differ from a vertical stretch/compression?

**Tip**: When sketching transformations, always start with basic shifts (vertical/horizontal) before applying stretches or reflections.

Use the graph of f to sketch transformations of the function, such as y = f(x) + 2, y = -f(x), y = f(x - 2), and others.

Learn how to apply vertical and horizontal shifts, reflections, and stretching transformations to the graph of f(x). This math problem covers key graph transformation concepts, including shifting, stretching, and reflecting, making it suitable for students in grades 9-12.

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