Describe the sequence and types of transformations required to transform the graph of y=x^3 into the graph of y=2f{-2(x-1)}+4. Give the equation of the new transformed function with detailed explanation.

To transform the graph of y = x^3 into the graph of y = 2f{-2(x-1)}+4, let's break down the sequence and types of transformations step by step:

1. Horizontal Translation:
The transformation -2(x-1) inside the function denotes a horizontal translation of 1 unit to the right (opposite sign of the inside function). So, the x-values are shifted 1 unit to the right.

2. Vertical Reflection:
The function inside f{} brackets (-2(x-1)) represents a vertical reflection across the x-axis (due to the negative coefficient). This means that the entire graph is flipped vertically.

3. Vertical Stretch:
The coefficient 2 in front of the transformed function f{} implies a vertical stretch by a factor of 2. This means that every y-value in the graph will be multiplied by 2.

4. Vertical Translation:
The constant term 4 at the end of the transformed function indicates a vertical translation of 4 units upwards. So, the graph is shifted 4 units up.

Combining these transformations, we can rewrite the equation step by step:

1. The graph of y = x^3 is shifted 1 unit to the right: y = (x-1)^3.

2. The graph is vertically reflected across the x-axis: y = -(x-1)^3.

3. The graph is vertically stretched by a factor of 2: y = -2(x-1)^3.

4. Finally, the graph is shifted 4 units up: y = -2(x-1)^3 + 4.

This is the equation of the new transformed function, which describes the graph of y = 2f{-2(x-1)}+4.