Let's analyze the transformation step-by-step to identify the graph of [tex]\( g(x) = (x-1)^3 + 4 \)[/tex].
1. Parent Function: The parent function is [tex]\( f(x) = x^3 \)[/tex]. This is a cubic function with no transformations applied to it. The basic shape of the graph is an S-curve that passes through the origin (0,0) and has the following properties:
- It is symmetric with respect to the origin.
- It passes through the points (1,1), (-1,-1), (2,8), and (-2,-8).
2. Horizontal Transformation: The function [tex]\( g(x) = (x-1)^3 + 4 \)[/tex] includes a horizontal transformation. The term [tex]\( (x - 1) \)[/tex] means we shift the graph of the parent function [tex]\( f(x) \)[/tex] to the right by 1 unit. Horizontal transformations are of the form [tex]\( f(x - h) \)[/tex], which shifts the graph to the right by [tex]\( h \)[/tex] units. In this case, [tex]\( h = 1 \)[/tex].
3. Vertical Transformation: In the function [tex]\( g(x) = (x-1)^3 + 4 \)[/tex], the [tex]\( + 4 \)[/tex] outside the cubic term represents a vertical transformation. This transformation shifts the graph upward by 4 units. Vertical transformations are of the form [tex]\( f(x) + k \)[/tex], which shifts the graph up by [tex]\( k \)[/tex] units. In this case, [tex]\( k = 4 \)[/tex].
By combining both transformations, we understand the following about [tex]\( g(x) \)[/tex]:
- The graph is the same shape as [tex]\( f(x) = x^3 \)[/tex], but it is shifted 1 unit to the right and 4 units up.
4. Graph Identification:
- The original critical point (0,0) of [tex]\( f(x) \)[/tex] will now be at (1,4) after the horizontal and vertical shifts.
- Other critical points will also shift accordingly:
- (1, 1) on [tex]\( f(x) \)[/tex] will become (2, 5) on [tex]\( g(x) \)[/tex].
- (-1, -1) on [tex]\( f(x) \)[/tex] will become (0, 3) on [tex]\( g(x) \)[/tex].
- (2, 8) on [tex]\( f(x) \)[/tex] will become (3, 12) on [tex]\( g(x) \)[/tex].
- (-2, -8) on [tex]\( f(x) \)[/tex] will become (-1, -4) on [tex]\( g(x) \)[/tex].
Thus, the graph of [tex]\( g(x) = (x-1)^3 + 4 \)[/tex] will look like a cubic function shifted to the right by 1 unit and up by 4 units. It will still retain the characteristic S-curve shape but with the new set of points reflecting the described transformations.
