How do you describe the transformation of #f(x)=(x-1)^3+2# from a common function that occurs and sketch the graph?

Question

Sadie Ackerman · Accepted Answer

a translation of #x^3# by the vector
#((1),(2))# In general he function #f(x+a)+b# is a translation of #f(x)# by the vector#((-a),(b))# for #f(x)=(x-1)^3+2# is a translation of #x^3# by the vector #((1),(2))# graph{(x-1)^3+2 [-10, 10, -5, 5]}

Sadie Ackerman · Answer

undefined The transformation of the function $ f(x) = (x - 1)^3 + 2 $ involves shifting the graph of the common cubic function $ y = x^3 $ to the right by 1 unit and upward by 2 units.

Sketching the graph:
1. Start with the common cubic function $ y = x^3 $.
2. Shift the graph to the right by 1 unit. This means each point on the graph moves 1 unit to the right.
3. Shift the resulting graph upward by 2 units. This means each point on the graph moves 2 units upward.
4. The final graph represents the transformed function $ f(x) = (x - 1)^3 + 2 $.

This transformation results in a graph that is similar in shape to the common cubic function but shifted to the right and upward.

Lincoln Anderson · Answer

undefined The function $ f(x) = (x - 1)^3 + 2 $ is a transformation of the basic cubic function $ f(x) = x^3 $. It has been shifted one unit to the right and two units up compared to the basic cubic function. To sketch the graph, you would start with the basic cubic function $ f(x) = x^3 $ and apply the transformations. So, the graph will still retain the general shape of a cubic function but will be shifted one unit to the right and two units up from the origin.