In this image, we are asked to find the formula for $g(x)$ in terms of $f(x)$, where $f$ is the blue graph on the left, and $g$ is the red graph on the right.

Step-by-step Analysis:

1. Look at the shape of the graphs: Both graphs represent piecewise linear functions. $f(x)$ has a sharp "V" shape centered at $(0, 0)$, and $g(x)$ appears to be a vertically shifted and slightly transformed version of $f(x)$.

2. Check for shifts or transformations:

   • The graph of $f(x)$ has vertices at $(-2, -4)$, $(0, 0)$, and $(2, -4)$. It is symmetric about the y-axis.
   • The graph of $g(x)$ has the same vertex pattern, but the function appears to be shifted down by 2 units compared to $f(x)$.

3. Conclude the transformation: From the observation that $g(x)$ looks exactly like $f(x)$ but shifted downward by 2 units, we can hypothesize: $g(x) = f(x) - 2$ This transformation corresponds to a vertical shift of the graph of $f(x)$ down by 2 units.

Final Answer:

$g(x) = f(x) - 2$

This is the formula for $g$ in terms of $f$.

Would you like further explanation, or do you have any specific questions?

Here are 5 related questions to expand your understanding:

1. How can you identify horizontal transformations from graphs?
2. What are other types of transformations, such as stretching or reflection?
3. How does shifting a function horizontally differ from vertically?
4. Can you express $f(x)$ in terms of a piecewise function based on its graph?
5. How would the formula for $g(x)$ change if the graph of $g(x)$ was a reflection of $f(x)$?

Tip: Always check for both vertical and horizontal shifts when comparing two similar graphs.