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: Analyze Parent Function

The parent function is \( f(x) = x^3 \), which has a point at \((0,0)\) (the inflection point).

Step2: Analyze Transformations

For \( g(x) = (x - 1)^3 + 2 \), the transformation rules are:

• Horizontal shift: \( (x - h) \) shifts the graph \( h \) units to the right. Here, \( h = 1 \), so shift right 1 unit.
• Vertical shift: \( + k \) shifts the graph \( k \) units up. Here, \( k = 2 \), so shift up 2 units.

Step3: Find Key Point of \( g(x) \)

Applying the shifts to the parent function's key point \((0,0)\):

• Shift right 1: \( x \)-coordinate becomes \( 0 + 1 = 1 \).
• Shift up 2: \( y \)-coordinate becomes \( 0 + 2 = 2 \). So the key point of \( g(x) \) is \((1, 2)\).

Step4: Compare with Given Graph

The given graph has its inflection point near \((0,0)\) (or \((0, -1)\) maybe, but not \((1,2)\)). Wait, maybe the other graphs (not fully shown, but assuming standard cubic shifts) – the correct graph should have the inflection point at \((1, 2)\). So we need to identify the graph with the inflection point at \((1, 2)\). Since the current graph shown has inflection at around \((0,0)\) (or \((0, -1)\)), the correct graph is the one shifted right 1 and up 2, so its inflection point is \((1, 2)\).

Answer:

The graph with the inflection point at \((1, 2)\) (assuming standard options, the correct graph would be the one shifted right 1 and up 2 from \( f(x) = x^3 \)). If we consider the options, the correct graph is the one where the cubic curve has its "center" at \((1, 2)\) instead of \((0,0)\).