https://file.aiquickdraw.com/mathbot/file/1728958873723tuigvfa0.jpg

It appears that you uploaded the same image again. Based on my previous explanation and after re-examining the question in the image, the transformations for \( h(x) = \frac{1}{5}(x - 7)^3 + 6 \) are still the same:

- **Horizontal Shift**: Right by 7 units (due to \( x - 7 \)).
- **Vertical Shrinking**: Shrunk by a factor of \( \frac{1}{5} \) (because of the \( \frac{1}{5} \) coefficient).
- **Vertical Shift**: Upward by 6 units (because of the +6).

Thus, the correct answer is **B**: "The graph of \( f(x) = x^3 \) should be horizontally shifted to the right by 7 units, shrunk vertically by a factor of \( \frac{1}{5} \), and shifted vertically up by 6 units."

Would you like more details or have further questions?

### Follow-up Questions:
1. What is the difference between horizontal and vertical shifts in function transformations?
2. Why does multiplying a function by a number less than 1 result in a vertical compression?
3. How would the transformation change if the function had \( +7 \) instead of \( -7 \) inside the parentheses?
4. Can you explain how these transformations would affect the shape of the graph visually?
5. How would the graph change if the function were \( -\frac{1}{5}(x - 7)^3 + 6 \)?

### Tip:
Always remember that horizontal transformations (inside the parentheses) affect the x-direction in a counterintuitive way: \( x - a \) shifts right, while \( x + a \) shifts left.

Use transformations of the standard cubic function, f(x) = x^3, to graph the function h(x) = 1/5 * (x - 7)^3 + 6. What transformations are needed?

Learn how to apply transformations to the cubic function f(x) = x^3 to graph h(x) = 1/5 * (x - 7)^3 + 6. This problem covers important concepts such as horizontal and vertical shifts, vertical compression, and transformations of cubic functions. Suitable for students in Grades 10-12.

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