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Answer :
To solve this problem, we need to determine how the function [tex]\( g(x) = (x+4)^3 \)[/tex] is derived from the function [tex]\( f(x) = x^3 \)[/tex].
1. Understand the Basic Function:
Our starting function is [tex]\( f(x) = x^3 \)[/tex]. This is a basic cubic function, and its graph passes through the origin (0, 0) and has a characteristic S-shape.
2. Identify the Transformation in [tex]\( g(x) = (x+4)^3 \)[/tex]:
Notice that [tex]\( g(x) \)[/tex] takes the form of [tex]\( f(x) \)[/tex] with [tex]\( x \)[/tex] replaced by [tex]\( (x+4) \)[/tex]. This is a transformed version of the function [tex]\( f(x) \)[/tex].
3. Horizontal Transformation:
The expression [tex]\((x+4)\)[/tex] indicates a horizontal transformation. Typically, [tex]\( f(x+c) \)[/tex] results in a horizontal shift of the graph of the function [tex]\( f(x) \)[/tex] by [tex]\( c \)[/tex] units. In this case, [tex]\( +4 \)[/tex] inside the function suggests a shift to the left by 4 units.
4. Conclusion:
Therefore, the graph of [tex]\( g(x) = (x+4)^3 \)[/tex] compared to [tex]\( f(x) = x^3 \)[/tex] is shifted 4 units to the left.
The correct description for the transformation is:
- A. a horizontal transformation of function [tex]\( f \)[/tex] 4 units left.
1. Understand the Basic Function:
Our starting function is [tex]\( f(x) = x^3 \)[/tex]. This is a basic cubic function, and its graph passes through the origin (0, 0) and has a characteristic S-shape.
2. Identify the Transformation in [tex]\( g(x) = (x+4)^3 \)[/tex]:
Notice that [tex]\( g(x) \)[/tex] takes the form of [tex]\( f(x) \)[/tex] with [tex]\( x \)[/tex] replaced by [tex]\( (x+4) \)[/tex]. This is a transformed version of the function [tex]\( f(x) \)[/tex].
3. Horizontal Transformation:
The expression [tex]\((x+4)\)[/tex] indicates a horizontal transformation. Typically, [tex]\( f(x+c) \)[/tex] results in a horizontal shift of the graph of the function [tex]\( f(x) \)[/tex] by [tex]\( c \)[/tex] units. In this case, [tex]\( +4 \)[/tex] inside the function suggests a shift to the left by 4 units.
4. Conclusion:
Therefore, the graph of [tex]\( g(x) = (x+4)^3 \)[/tex] compared to [tex]\( f(x) = x^3 \)[/tex] is shifted 4 units to the left.
The correct description for the transformation is:
- A. a horizontal transformation of function [tex]\( f \)[/tex] 4 units left.
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