Sure, let's break down the transformation of the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] into the new function [tex]\( g(x) = 2f(x-3) \)[/tex] step-by-step.
1. Understanding the Parent Function:
The parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is a cube root function. The basic shape of this graph passes through the origin and has a characteristic S-shaped curve, which extends infinitely in both directions along the x-axis.
2. Horizontal Shift:
The term [tex]\( (x - 3) \)[/tex] inside the function indicates a horizontal shift. Specifically, since it is [tex]\( x - 3 \)[/tex], it means the graph will shift 3 units to the right.
- The graph of [tex]\( f(x - 3) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] moved 3 units rightwards.
3. Vertical Stretch:
The coefficient 2 outside the function indicates a vertical stretch by a factor of 2.
- Multiplying the function by 2 means every y-value on the original graph will be doubled, stretching the graph vertically.
So, combining both transformations:
- The graph is first shifted 3 units to the right.
- It is then stretched vertically by a factor of 2.
To summarize, the transformed function [tex]\( g(x) = 2\sqrt[3]{x - 3} \)[/tex] will be:
- Horizontally shifted 3 units to the right.
- Vertically stretched by a factor of 2.
Given this understanding, the graph of [tex]\( g(x) \)[/tex] will reflect these transformations. Consequently, the correct graph will exhibit:
- The same S-shaped curve characteristic of the cube root function.
- The curve moved to start from a point that is 3 units to the right of what it would for [tex]\( \sqrt[3]{x} \)[/tex].
- An elongated curve stretching vertically due to the factor of 2.
Please refer to the graphs provided to determine which one matches these characteristics. The graph showing these specific shifts and stretches is the correct one.
Given the transformations described, the correct answer is the graph that corresponds to these changes.
