To describe the transformation that maps the function [tex]\( f(x) = x^3 \)[/tex] onto [tex]\( f(x) = 2(x-5)^3 \)[/tex], let's break down the steps involving transformations.
1. Horizontal Shift:
- The function [tex]\( f(x) = x^3 \)[/tex] is transformed into [tex]\( f(x) = (x-5)^3 \)[/tex].
- The term [tex]\( (x-5) \)[/tex] implies a horizontal shift to the right by 5 units. This is because subtracting 5 from [tex]\( x \)[/tex] moves the graph to the right.
2. Vertical Stretch:
- The function [tex]\( f(x) = (x-5)^3 \)[/tex] is then transformed into [tex]\( f(x) = 2(x-5)^3 \)[/tex].
- Multiplying the entire function by 2 means that every [tex]\( y \)[/tex]-value is scaled by a factor of 2. This transformation is called a vertical stretch by a factor of 2.
Thus, combining these two transformations, you get:
- A horizontal shift to the right by 5 units.
- A vertical stretch by a factor of 2.
Therefore, the correct description of the transformations is:
Shift right 5; vertical stretch by a factor of 2.
