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To determine the graph of the function [tex]\( g(x) = (x-1)^3 + 1 \)[/tex], we need to understand how translations affect the graph of the parent function [tex]\( f(x) = x^3 \)[/tex].
1. Horizontal Translation:
- The term [tex]\( x-1 \)[/tex] inside the cubic function indicates a horizontal translation.
- Specifically, the graph of [tex]\( f(x) = x^3 \)[/tex] is shifted to the right by 1 unit. This is because [tex]\( x \)[/tex] is replaced by [tex]\( x-1 \)[/tex], which moves the graph rightward.
2. Vertical Translation:
- The constant [tex]\( +1 \)[/tex] outside the cubic function indicates a vertical translation.
- This means that the entire graph is shifted upward by 1 unit.
Putting these translations together:
1. Original Function: The original function [tex]\( f(x) = x^3 \)[/tex] passes through the origin (0,0) and has key points such as [tex]\((1,1)\)[/tex] and [tex]\((-1,-1)\)[/tex].
2. Horizontal Shift:
- Shifting the graph 1 unit to the right changes the coordinates of any point [tex]\((x, y)\)[/tex] on [tex]\( f(x) \)[/tex] to [tex]\((x+1, y)\)[/tex].
- For example, the origin [tex]\((0,0)\)[/tex] becomes [tex]\((1,0)\)[/tex], the point [tex]\((1,1)\)[/tex] becomes [tex]\((2,1)\)[/tex], and [tex]\((-1,-1)\)[/tex] becomes [tex]\((0,-1)\)[/tex].
3. Vertical Shift:
- Shifting the graph 1 unit upward changes the coordinates of any point [tex]\((x, y)\)[/tex] on the horizontally shifted graph to [tex]\((x, y+1)\)[/tex].
- Continuing with our points: [tex]\((1,0)\)[/tex] becomes [tex]\((1,1)\)[/tex], [tex]\((2,1)\)[/tex] becomes [tex]\((2,2)\)[/tex], and [tex]\((0,-1)\)[/tex] becomes [tex]\((0,0)\)[/tex].
Thus, after applying both translations, the graph of [tex]\( g(x) = (x-1)^3 + 1 \)[/tex] will maintain the same general shape as the graph of [tex]\( f(x) = x^3 \)[/tex], but shifted to the right by 1 unit and up by 1 unit. This results in new key points:
- [tex]\((1,1)\)[/tex] (formerly the origin)
- [tex]\((2,2)\)[/tex] (formerly [tex]\((1,1)\)[/tex])
- [tex]\((0,0)\)[/tex] (formerly [tex]\((-1,-1)\)[/tex])
The graph demonstrably passes through the point [tex]\((2,2)\)[/tex] validating the upward translation. The transformation confirms the expected rightward shift culminating the point [tex]\((1,1)\)[/tex]. Notice, the graph between the former points [tex]\((0,0)\)[/tex] portrays an intact [tex]\( x^3 \)[/tex] reflecting translational integrity .
Conclusively, the defined transformation matches a translated function graphically represented by rightward and upward shifts infused. Aforementioned transformations inherently elucidate a shift enveloping total resulting coordinative shifts locally defining new translatory local points through the cubic function [tex]\( g(x) = (x-1)^3 + 1 \)[/tex].
1. Horizontal Translation:
- The term [tex]\( x-1 \)[/tex] inside the cubic function indicates a horizontal translation.
- Specifically, the graph of [tex]\( f(x) = x^3 \)[/tex] is shifted to the right by 1 unit. This is because [tex]\( x \)[/tex] is replaced by [tex]\( x-1 \)[/tex], which moves the graph rightward.
2. Vertical Translation:
- The constant [tex]\( +1 \)[/tex] outside the cubic function indicates a vertical translation.
- This means that the entire graph is shifted upward by 1 unit.
Putting these translations together:
1. Original Function: The original function [tex]\( f(x) = x^3 \)[/tex] passes through the origin (0,0) and has key points such as [tex]\((1,1)\)[/tex] and [tex]\((-1,-1)\)[/tex].
2. Horizontal Shift:
- Shifting the graph 1 unit to the right changes the coordinates of any point [tex]\((x, y)\)[/tex] on [tex]\( f(x) \)[/tex] to [tex]\((x+1, y)\)[/tex].
- For example, the origin [tex]\((0,0)\)[/tex] becomes [tex]\((1,0)\)[/tex], the point [tex]\((1,1)\)[/tex] becomes [tex]\((2,1)\)[/tex], and [tex]\((-1,-1)\)[/tex] becomes [tex]\((0,-1)\)[/tex].
3. Vertical Shift:
- Shifting the graph 1 unit upward changes the coordinates of any point [tex]\((x, y)\)[/tex] on the horizontally shifted graph to [tex]\((x, y+1)\)[/tex].
- Continuing with our points: [tex]\((1,0)\)[/tex] becomes [tex]\((1,1)\)[/tex], [tex]\((2,1)\)[/tex] becomes [tex]\((2,2)\)[/tex], and [tex]\((0,-1)\)[/tex] becomes [tex]\((0,0)\)[/tex].
Thus, after applying both translations, the graph of [tex]\( g(x) = (x-1)^3 + 1 \)[/tex] will maintain the same general shape as the graph of [tex]\( f(x) = x^3 \)[/tex], but shifted to the right by 1 unit and up by 1 unit. This results in new key points:
- [tex]\((1,1)\)[/tex] (formerly the origin)
- [tex]\((2,2)\)[/tex] (formerly [tex]\((1,1)\)[/tex])
- [tex]\((0,0)\)[/tex] (formerly [tex]\((-1,-1)\)[/tex])
The graph demonstrably passes through the point [tex]\((2,2)\)[/tex] validating the upward translation. The transformation confirms the expected rightward shift culminating the point [tex]\((1,1)\)[/tex]. Notice, the graph between the former points [tex]\((0,0)\)[/tex] portrays an intact [tex]\( x^3 \)[/tex] reflecting translational integrity .
Conclusively, the defined transformation matches a translated function graphically represented by rightward and upward shifts infused. Aforementioned transformations inherently elucidate a shift enveloping total resulting coordinative shifts locally defining new translatory local points through the cubic function [tex]\( g(x) = (x-1)^3 + 1 \)[/tex].
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