Let's analyze the problem:

The functions provided are:

1. $f(x) = x^2$
2. $g(x) = x^2 - 8$

The question asks for the transformation that describes how $g(x)$ relates to $f(x)$.

Solution

The function $g(x) = x^2 - 8$ is derived from $f(x) = x^2$ by subtracting 8 from the function's output. This subtraction of 8 represents a vertical shift downward by 8 units.

Thus, the correct transformation from $f(x)$ to $g(x)$ is a vertical shift down by 8 units.

Answer

The correct option is:

• D) Function $g$ shifted down eight units from function $f$

Would you like further clarification or have any questions?

Here are 5 related questions that build upon this concept:

1. How do vertical and horizontal shifts differ in function transformations?
2. What effect would adding 8 to $f(x) = x^2$ have on its graph?
3. How does multiplying $f(x) = x^2$ by a constant affect its steepness?
4. How would $g(x) = (x-8)^2$ differ in transformation from $f(x) = x^2$?
5. Can transformations involve both shifting and stretching a function?

Tip: When analyzing transformations, always pay attention to whether changes are applied inside the function (affecting x) or outside (affecting y) to determine the direction and type of shift.