The problem states:

Function $g$ is defined by $g(x) = \sqrt{x}$, and function $f$ is defined by $f(x) = \sqrt{x + 4}$. In the $xy$-plane, how does the graph of $y = g(x)$ compare to the graph of $y = f(x)$?

To analyze this, we need to determine how the graph of $f(x) = \sqrt{x + 4}$ relates to the graph of $g(x) = \sqrt{x}$.

Translation Analysis:

The graph of $f(x) = \sqrt{x + 4}$ can be considered as a horizontal translation of the graph of $g(x) = \sqrt{x}$. The expression $\sqrt{x + 4}$ means that the graph of $\sqrt{x}$ is shifted horizontally to the left by 4 units.

Conclusion:

• The correct answer is option D: "The graph of $y = f(x)$ is a translation of the graph of $y = g(x)$ four units to the left."

Let me know if you would like further details or have any other questions! Here are 5 related questions you may find useful:

1. How do you determine if a graph is translated horizontally or vertically?
2. What happens to the graph of $y = \sqrt{x}$ if we subtract a constant from $x$?
3. How would the graph change if the function was $\sqrt{x - 4}$ instead of $\sqrt{x + 4}$?
4. Can translations be combined with other transformations like reflections or scaling?
5. How would the transformation differ for a function like $y = x^2$?

Tip: Always remember that for horizontal translations, the shift is opposite to the sign inside the function: $f(x - h)$ shifts to the right by $h$ units, and $f(x + h)$ shifts to the left by $h$ units.