To analyze this transformation between $f(x)$ and $g(x)$, we will focus on understanding how the graph of $f(x)$ has been modified to create $g(x)$.

Graphical Analysis:

• The graph of $f(x)$ is shown in green, while $g(x)$ is in red.
• To figure out the transformation, we look for shifts (horizontal and vertical) and possibly reflections or scalings.

From the graph:

• $g(x)$ appears to be shifted downward compared to $f(x)$.
• There also appears to be a horizontal shift as $g(x)$ is shifted to the right.

Vertical and Horizontal Shifts:

1. Vertical Shift: Comparing the y-values, $g(x)$ seems to be shifted downward by 4 units compared to $f(x)$. This indicates a vertical transformation of $-4$.

2. Horizontal Shift: Comparing the x-values, $g(x)$ seems to be shifted to the right by 2 units compared to $f(x)$. This suggests a horizontal transformation involving $(x - 2)$.

Conclusion:

The correct transformation that defines $g(x)$ in terms of $f(x)$ is: $g(x) = f(x - 2) - 4$ This corresponds to the first option: $g(x) = f(x - 2) - 4$

Would you like further details or have any questions about this?

Here are 5 questions to expand on this topic:

1. How do horizontal shifts affect the equation of a function?
2. How can you determine vertical shifts by looking at a graph?
3. What is the effect of a transformation of the form $f(x + a)$?
4. Can a function's graph be transformed by reflections or scalings in addition to shifts?
5. How do transformations help in understanding function behavior in real-world applications?

Tip: To quickly identify transformations, compare key points of the function (like the intercepts or maxima/minima) before and after transformation!