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The questions you're asking about are 37 and 38, which are:

37. \( p(x) = \left(\frac{1}{3}x\right)^3 + 2 \)

38. \( p(x) = \left[3(x - 2)\right]^3 \)

You are asked to compare the graphs of these functions with the graph of \( f(x) = x^3 \).

### Let's break down each function:

#### 37. \( p(x) = \left(\frac{1}{3}x\right)^3 + 2 \)
- The term \( \left(\frac{1}{3}x\right)^3 \) represents a horizontal stretch of the parent function \( f(x) = x^3 \) by a factor of 3. This stretches the graph horizontally, making it wider.
- The \( +2 \) shifts the entire graph vertically up by 2 units.

Thus, the graph of \( p(x) \) is a horizontally stretched version of \( f(x) = x^3 \), shifted upward by 2 units.

#### 38. \( p(x) = \left[3(x - 2)\right]^3 \)
- The term \( 3(x - 2) \) inside the cubic function represents a horizontal compression by a factor of 1/3 (since the factor of 3 inside makes the graph narrower).
- The \( x - 2 \) translates the graph 2 units to the right.

Thus, the graph of this function is a horizontally compressed version of \( f(x) = x^3 \), shifted 2 units to the right.

### Summary of transformations:
- **37**: Horizontal stretch by a factor of 3, vertical shift up by 2 units.
- **38**: Horizontal compression by a factor of 1/3, horizontal shift to the right by 2 units.

Would you like further details or a graph for visual comparison?

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Here are 5 related questions for deeper exploration:
1. How would the graph of \( f(x) = (x + 1)^3 \) compare with \( f(x) = x^3 \)?
2. How does multiplying by a constant inside the function affect the graph's width?
3. What happens when a constant is added outside the function?
4. How does the order of transformations (stretch vs. shift) affect the final graph?
5. Can you describe how the graph of \( f(x) = -x^3 \) would look in comparison to \( x^3 \)?

**Tip**: Always apply horizontal transformations (shifts, stretches/compressions) before vertical transformations when analyzing function graphs.

37. p(x) = (1/3x)^3 + 2 and 38. p(x) = [3(x - 2)]^3. Compare the graphs with the graph of f(x) = x^3.

Explore the transformations of cubic functions p(x) = (1/3x)^3 + 2 and p(x) = [3(x - 2)]^3, including horizontal stretches, vertical shifts, and graph compression. Understand how these functions compare with the parent function f(x) = x^3. Suitable for high school math students in Grades 10-12.

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