The function $g(x) = (x + 4)^2$ is a transformation of the basic quadratic function $f(x) = x^2$.

To analyze this transformation, observe the following:

• The transformation of $f(x) = x^2$ into $g(x) = (x + 4)^2$ involves the expression inside the parentheses. The graph of $f(x)$ is shifted horizontally based on the value added or subtracted inside the parentheses.

• In general, for a function $f(x) = (x + h)^2$, the graph of $f(x)$ is shifted h units to the left if $h > 0$, or to the right if $h < 0$.

For the function $g(x) = (x + 4)^2$, the "+4" indicates a shift of 4 units to the left. This is because the $+4$ inside the parentheses shifts the graph to the left.

Thus, the correct transformation is a shift of 4 units to the left.

Answer: D. The graph of $f(x) = x^2$ should be shifted 4 units to the left.

Would you like further clarification or more details on how transformations work?

Here are 5 related questions you might find interesting:

1. What are the different types of transformations that can be applied to the graph of a function?
2. How does a horizontal shift differ from a vertical shift in function graphs?
3. How would the graph of $g(x) = (x - 4)^2$ differ from $f(x) = x^2$?
4. What happens to the graph of $f(x) = x^2$ if we multiply it by a negative constant, like $-x^2$?
5. Can transformations like shifts and reflections be combined on a single graph?

Tip: Remember that horizontal shifts are opposite to what you might expect based on the sign inside the parentheses: $(x + h)$ shifts to the left, while $(x - h)$ shifts to the right.