The question asks you to determine which transformations are needed to graph the function $g(x) = x^3 - 2$ using the base function $f(x) = x^3$.

The function $g(x) = x^3 - 2$ is a simple transformation of the base function $f(x) = x^3$. The transformation here involves a vertical shift. Specifically:

1. The term "-2" represents a vertical shift downward by 2 units.

Let's go through the choices:

• A. Shift the graph 2 units up: Incorrect, as the transformation shifts the graph down, not up.
• B. Stretch the graph vertically by a factor of 2: Incorrect, no vertical stretch is involved.
• C. Shift the graph 2 units to the right: Incorrect, no horizontal shift is involved.
• D. Shrink the graph vertically by a factor of 2: Incorrect, no vertical shrink occurs.
• E. Reflect the graph about the y-axis: Incorrect, no reflection is involved.
• F. Shrink the graph horizontally by a factor of 2: Incorrect, no horizontal shrink happens.
• G. Reflect the graph about the x-axis: Incorrect, the graph is not reflected.
• H. Shift the graph 2 units to the left: Incorrect, no horizontal shift is involved.
• I. Shift the graph 2 units down: Correct, the graph is shifted 2 units downward.
• J. Stretch the graph horizontally by a factor of 2: Incorrect, no horizontal stretching happens.

The correct answer is:

• I. Shift the graph 2 units down.

Would you like more details or have any questions?

Here are 5 related questions you might consider:

1. How do vertical and horizontal shifts affect the graph of a function?
2. What does adding or subtracting a constant from a function do to its graph?
3. How do you recognize reflections in function transformations?
4. How do you apply vertical or horizontal stretches and compressions?
5. What transformations would be needed for $g(x) = 2x^3$?

Tip: Remember that transformations involving shifts are indicated by adding or subtracting constants either inside the function for horizontal shifts or outside for vertical shifts.