To determine which graph represents the function [tex]\( g(x) = (x-1)^3 + 4 \)[/tex] based on the transformation of the parent function [tex]\( f(x) = x^3 \)[/tex], let's analyze the transformations step-by-step.
1. Horizontal Shift: The term [tex]\((x-1)^3\)[/tex] indicates a horizontal shift. Specifically, [tex]\( x-1 \)[/tex] means that the graph of [tex]\( f(x) = x^3 \)[/tex] is shifted to the right by 1 unit. This is because replacing [tex]\( x \)[/tex] with [tex]\( x-1 \)[/tex] translates the graph to the right.
2. Vertical Shift: The [tex]\( +4 \)[/tex] outside the parentheses means the graph is shifted vertically upwards by 4 units. This is because adding a constant to a function results in a vertical shift.
In summary, the original graph of [tex]\( f(x) = x^3 \)[/tex] undergoes two transformations:
- It is shifted to the right by 1 unit.
- It is shifted up by 4 units.
To identify the correct graph representation of [tex]\( g(x) \)[/tex]:
- Look for the graph where the similar cubic shape of [tex]\( x^3 \)[/tex] is present but relocated.
- The cubic graph should be clearly shifted 1 unit right and 4 units up from the standard position of [tex]\( x=0 \)[/tex], [tex]\( y=0 \)[/tex].
Graph A and Graph B should provide visual cues showing these transformations. You’ll select the graph that appropriately displays the function shifted as specified.
Based on the information we have:
"The graph of [tex]\( g(x) = (x-1)^3 + 4 \)[/tex] is shifted to the right by 1 unit and up by 4 units relative to [tex]\( f(x) = x^3 \)[/tex]."
Hence, you would examine the given graphs and choose the one that correctly demonstrates these specific transformations.
