The function you are asked to analyze is:

$f(x) = (x - 7)^3$

This is a cubic function that has undergone a horizontal shift. Here's a breakdown of the transformations:

1. Horizontal Shift: The function has been shifted to the right by 7 units, as indicated by the term $(x - 7)$. This is a transformation of the basic cubic function $f(x) = x^3$. The shift happens because the function takes the form $(x - h)$, where $h = 7$, meaning a shift to the right by 7 units.

2. Vertical Stretch/Compress: There is no vertical stretch or compression in this function because there are no coefficients multiplying $(x - 7)^3$.

3. Reflection: There are no negative signs in front of the function, so there is no reflection across either axis.

Answer Breakdown for the Options in the Tool:

• Horizontal Shift: Right
• Vertical Stretch/Compress: None
• x-Axis Reflection: No
• y-Axis Reflection: No

This should correctly represent the transformation of the cubic function.

Would you like more details on the steps, or do you have any questions about transformations?

Here are five follow-up questions to expand on this:

1. How do horizontal and vertical shifts affect the shape of a function?
2. How would a reflection across the x-axis change the appearance of a cubic function?
3. What does a vertical stretch or compression do to a graph?
4. If a function is reflected across both axes, how would its graph be impacted?
5. How can you determine the direction of a horizontal shift just by looking at the equation?

Tip: A horizontal shift occurs in the opposite direction of the sign inside the parentheses. For example, $f(x - 7)$ shifts the graph right by 7 units, not left.