https://file.aiquickdraw.com/mathbot/file/1725852552105s70jn015.jpg

https://file.aiquickdraw.com/mathbot/file/1725852552105iroqnfn4.jpg

The two images you've provided give us the graph of the function \( f(x) \) and a set of transformations that should be applied to it. Let's go through the transformations one by one.

### Given Function Graph:
The first image shows the graph of \( f(x) \), which seems to resemble a sinusoidal (sine or cosine) function with key points.

### Transformations and Their Effects:
1. **Problem 78: \( g(x) = f(x) - 2 \)**
   - This represents a **vertical translation** of the graph of \( f(x) \) by 2 units **down**. Every point on the graph of \( f(x) \) moves 2 units downward.

2. **Problem 79: \( g(x) = -f(x) \)**
   - This transformation represents a **reflection** across the x-axis. The entire graph of \( f(x) \) is flipped upside down. Points that were above the x-axis will move below it, and vice versa.

3. **Problem 80: \( g(x) = f(x + 1) \)**
   - This represents a **horizontal shift** of the graph of \( f(x) \) by 1 unit **to the left**. Every point on the graph shifts left by 1 unit.

4. **Problem 81: \( g(x) = f(x - 2) \)**
   - This represents a **horizontal shift** of the graph of \( f(x) \) by 2 units **to the right**. Every point on the graph shifts right by 2 units.

### Recap of the Transformations:
- **78**: Vertical shift down 2 units.
- **79**: Reflection over the x-axis.
- **80**: Horizontal shift left by 1 unit.
- **81**: Horizontal shift right by 2 units.

Would you like more detailed steps for graphing these transformations, or do you have further questions about any of these?

Here are five related questions to deepen your understanding:
1. How does a vertical shift affect the maximum and minimum points of a function?
2. What happens to the x-intercepts of the function after a reflection over the x-axis?
3. How does a horizontal shift affect the period and frequency of a sinusoidal function like \( f(x) \)?
4. Can horizontal and vertical shifts be combined into a single transformation?
5. How do these transformations apply to non-periodic functions like polynomials?

**Tip:** When dealing with transformations, always keep track of whether they affect the x-values (horizontal changes) or y-values (vertical changes) of the function.

Transform the function f(x) based on the following equations: g(x) = f(x) - 2, g(x) = -f(x), g(x) = f(x + 1), g(x) = f(x - 2).

Learn how to apply transformations to the function f(x), including vertical translations, reflections over the x-axis, and horizontal shifts. Understand key function transformation concepts and graphing techniques. Suitable for students in Grades 9-12.

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