The problem is asking to identify the value of $k$ that transforms the function $f(x)$ into $g(x) = f(x) + k$. This transformation represents a vertical shift of the graph of $f(x)$ by $k$ units.

Steps to Solve:

1. Observe the graph:

   • Identify corresponding points on $f(x)$ and $g(x)$.
   • For example, look at where $f(x)$ and $g(x)$ intersect the same $x$-coordinate.

2. Calculate the vertical shift $k$:

   • Subtract the $y$-coordinate of $f(x)$ from the $y$-coordinate of $g(x)$ at a chosen $x$-coordinate.
   • For instance, if $f(x)$ passes through $(-2, -2)$ and $g(x)$ passes through $(-2, 2)$, then: $k = y_{\text{g}} - y_{\text{f}} = 2 - (-2) = 4$

3. Generalize:

   • The value of $k$ is consistent across all points since it is a vertical shift.

Thus, in this problem, $k$ is 4.

Would you like further explanation or have additional questions? Here are 5 questions to extend your understanding:

1. How does the transformation $g(x) = f(x) + k$ affect the shape of the graph of $f(x)$?
2. What is the difference between a vertical and a horizontal shift of a function?
3. Can $k$ ever be negative in $g(x) = f(x) + k$? What would that mean graphically?
4. How can you identify $k$ using points not shown explicitly on the axes?
5. How would the transformation change if the function were $g(x) = f(x - h) + k$?

Tip: Always compare the $y$-coordinates at the same $x$-coordinate when finding vertical shifts.