To determine how the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] has been shifted to form the translation, let's analyze each translation option:
1. 2 units up: If the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is shifted 2 units up, this translates to moving every point on the graph vertically upwards by 2 units. Mathematically, this can be represented as the function:
[tex]\[
g(x) = \sqrt[3]{x} + 2
\][/tex]
In this scenario, each point [tex]\((x, f(x))\)[/tex] on the original graph will be transformed to [tex]\((x, f(x) + 2)\)[/tex].
2. 2 units down: If the graph is shifted 2 units down, every point on the graph moves vertically downwards by 2 units. This translation can be represented by the function:
[tex]\[
g(x) = \sqrt[3]{x} - 2
\][/tex]
Here, each point [tex]\((x, f(x))\)[/tex] on the graph of [tex]\( f(x) \)[/tex] would transform to [tex]\((x, f(x) - 2)\)[/tex].
3. 2 units left: Shifting the graph 2 units to the left means moving every point horizontally to the left by 2 units. Mathematically, this translates to:
[tex]\[
g(x) = \sqrt[3]{x + 2}
\][/tex]
In this case, each point [tex]\((x, f(x))\)[/tex] on the original graph would become [tex]\((x - 2, f(x))\)[/tex].
4. 2 units right: Shifting the graph 2 units to the right implies moving every point horizontally to the right by 2 units, which can be represented by:
[tex]\[
g(x) = \sqrt[3]{x - 2}
\][/tex]
Here, each point [tex]\((x, f(x))\)[/tex] on the graph of [tex]\( f(x) \)[/tex] would transform to [tex]\((x + 2, f(x))\)[/tex].
Given the result from the translation, which is [tex]\( (0, 2) \)[/tex], we can conclude that the correct transformation involves shifting the graph 2 units up. Therefore, the answer is:
2 units up
