Question

the graph of the parent function $f(x) = x^3$ is translated to form the graph of $g(x) = (x - 1)^3 + 2$. which is the graph of $g(x)$, the translated function?

Explanation:

Step1: Identify parent function vertex

The parent function $f(x)=x^3$ has a point of inflection (vertex-like point) at $(0,0)$.

Step2: Apply horizontal translation

For $g(x)=(x-1)^3+2$, the $(x-1)$ term shifts the graph 1 unit right. The new x-coordinate of the inflection point is $0+1=1$.

Step3: Apply vertical translation

The $+2$ term shifts the graph 2 units up. The new y-coordinate of the inflection point is $0+2=2$. So the inflection point of $g(x)$ is $(1,2)$.

Step4: Match to the graph

Locate the graph where the cubic curve's inflection point is at $(1,2)$, and the curve follows the shape of $x^3$ (increasing, S-shaped) shifted right 1 and up 2. This corresponds to the topmost graph (Option A).

Answer:

A. The topmost graph with inflection point at (1, 2)