https://file.aiquickdraw.com/mathbot/file/1727652345027t2bedu7k.jpg

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.

Consider the following function: f(x) = (x - 7)^3. Graph the original function by indicating how the more basic function has been shifted, reflected, stretched, or compressed.

Learn how to graph a transformed cubic function f(x) = (x - 7)^3 by applying horizontal shifts, without any reflections or vertical stretches. Understand key concepts in algebra for high school students.

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