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.