In this problem, we are given two graphs, one for $f(x)$ (the blue curve) and one for $g(x)$ (the red curve). We are asked to find the formula for $g(x)$ in terms of $f(x)$.

Observations:

• The graph of $f(x)$ appears to be a cubic or higher-degree function based on its oscillating nature.
• The graph of $g(x)$ appears to resemble a horizontally shifted version of $f(x)$.

Hypothesis:

By visually comparing the graphs, it looks like $g(x)$ is a horizontal shift of $f(x)$. More specifically:

• The local maximum and minimum points of $f(x)$ seem to shift to the right in $g(x)$.
• Therefore, the transformation might be of the form: $g(x) = f(x - c)$ where $c$ is the horizontal shift.

Estimating the Shift:

From the graphs:

• $f(x)$ has a local minimum around $x = -2$ and $x = 1$.
• $g(x)$ has its corresponding minimum at $x = 0$ and $x = 3$, indicating a shift of approximately $+2$ units.

Thus, a likely formula for $g(x)$ in terms of $f(x)$ is: $g(x) = f(x - 2)$

Would you like to explore this further or check specific points to verify?

Related Questions:

1. How does a horizontal shift affect the graph of a function?
2. What are the key differences between vertical and horizontal shifts?
3. Can we generalize this shift to any function $f(x)$?
4. What happens to the domain and range of a function after a horizontal shift?
5. Can you express other transformations, such as reflections or dilations, similarly?

Tip: For horizontal shifts, remember that shifting right corresponds to subtracting from $x$, while shifting left corresponds to adding to $x$.