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The problem describes a function and several transformations that need to be applied to it in sequence. Let's break down the problem step by step:

1. **Given Function:**
   \[
   f(x) = x^3
   \]

2. **Transformations to be applied (in order):**
   - **Shift left 2:** This shifts the graph of the function horizontally to the left by 2 units. The transformed function becomes:
     \[
     f(x) = (x + 2)^3
     \]
   
   - **Shift down 4:** This shifts the graph vertically downward by 4 units. The function becomes:
     \[
     f(x) = (x + 2)^3 - 4
     \]

   - **Dilate horizontally by a factor of 5:** This means the function is stretched horizontally by multiplying the input \( x \) by \(\frac{1}{5}\). The function now becomes:
     \[
     f(x) = \left(\frac{x + 2}{5}\right)^3 - 4
     \]

   - **Reflect over the x-axis:** This reflection inverts the graph over the x-axis by multiplying the entire function by \(-1\). The final transformed function is:
     \[
     f(x) = -\left(\frac{x + 2}{5}\right)^3 + 4
     \]

**Final Transformed Function:**
\[
f(x) = -\left(\frac{x + 2}{5}\right)^3 + 4
\]

### Second Part of the Problem
It asks to identify the parent function and describe the transformations for a different function:
\[
f(x) = -3|x - 3| + 2
\]
1. **Parent Function:** The parent function is the absolute value function, \( f(x) = |x| \).

2. **Transformations:**
   - **Reflect over x-axis:** Multiply by \(-1\), giving \( f(x) = -|x| \).
   - **Vertical stretch by 3:** Multiply by \(3\), giving \( f(x) = -3|x| \).
   - **Shift right 3:** Subtract 3 from \( x \), giving \( f(x) = -3|x - 3| \).
   - **Shift up 2:** Add 2, giving the final transformed function:
     \[
     f(x) = -3|x - 3| + 2
     \]

Would you like more details or have any questions?

**Related Questions:**
1. What is the effect of a horizontal dilation on the shape of the graph?
2. How does reflecting a function over the x-axis affect its graph?
3. What are the general rules for horizontal and vertical translations of functions?
4. How does a vertical stretch differ from a horizontal stretch?
5. Can you describe how to graph \( f(x) = -3|x| + 2 \)?

**Tip:** Always apply transformations in the correct order to avoid errors in the final function form.

f(x) = x^3, shift left 2, down 4, dilate horizontally by a factor of 5, reflect over the x-axis. Write the equation for the final transformed graph.

Learn how to apply multiple transformations, including horizontal shifts, vertical shifts, horizontal dilations, and reflections, to the cubic function f(x) = x^3. This detailed step-by-step solution explains how these transformations affect the graph of the function.

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