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 new transformed function with detailed explanation.
To transform the graph of y = x^3 into the graph of y = 2f[-2(x-1)] + 4, we need to go through several transformations:
1. Horizontal shift to the right by 1 unit:
We start by shifting the graph of y = x^3 one unit to the right. To do this, we replace 'x' with '(x-1)' in the equation. This gives us y = (x-1)^3.
2. Horizontal compression by a factor of 2:
Next, we compress the graph horizontally by a factor of 2. To achieve this, we replace 'x' with '2(x-1)' in the equation. This gives us y = (2(x-1)-1)^3. Simplifying, we get y = (2x - 3)^3.
3. Vertical stretch by a factor of 2:
To stretch the graph vertically by a factor of 2, we multiply the whole equation by 2. This gives us 2y = (2x - 3)^3.
4. Vertical shift upwards by 4 units:
Finally, we shift the graph vertically upwards by 4 units. We add 4 to the equation, resulting in 2y + 4 = (2x - 3)^3.
However, instead of using 'y', the transformed equation uses 'f[-2(x-1)]'. This means that the y-values are replaced by f[-2(x-1)]. Therefore, the final transformed equation becomes:
2f[-2(x-1)] + 4 = (2x - 3)^3.
This equation represents the graph of y = 2f[-2(x-1)] + 4, which is the transformed graph of y = x^3.