Original function f(x) = x^2 shows an upward u shape with points (0,0), (1,1), (-1,1), (2,4),(-2,4)

The original function is \( f(x) = x^2 \), and the translated function is \( f(x) = (x+1)^2 \). The translation from \( f(x) \) to \( f(x+k) \) indicates a horizontal shift.

In general, a horizontal shift to the left by \( k \) units is represented by \( f(x+k) \), which implies a substitution \( x \) with \( x+k \).
For the function \( f(x) = (x + 1)^2 \), we can rewrite it in the form of the original function:

\[
f(x) = f(x + 1)
\]

This tells us that the graph of \( f(x) = x^2 \) was shifted to the left by 1 unit to obtain the graph of \( f(x) = (x + 1)^2 \).

Thus, we can determine that \( k = -1 \) for the horizontal shift that occurred. The negative sign indicates a leftward shift.

Therefore, the value of \( k \) is:

\[
\boxed{-1}
\]