To determine the transformation of the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] into [tex]\( g(x) = f(x+2) - 4 \)[/tex], follow these steps:
1. Understand the parent function: The function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] represents a cubic root function. This graph is symmetric about the origin and increases from left to right, passing through the points [tex]\((-1, -1)\)[/tex], [tex]\((0, 0)\)[/tex], and [tex]\((1, 1)\)[/tex].
2. Analyze the transformation [tex]\( f(x+2) \)[/tex]: The expression [tex]\( f(x+2) \)[/tex] indicates a horizontal shift. Specifically, adding 2 inside the function argument [tex]\( x \)[/tex] (i.e., [tex]\( x+2 \)[/tex]) shifts the graph to the left by 2 units.
- Original point [tex]\((x, f(x))\)[/tex] becomes [tex]\((x, f(x+2))\)[/tex].
- For example, the point [tex]\((0, 0)\)[/tex] moves to [tex]\((0-2, 0)\)[/tex], which is [tex]\((-2, 0)\)[/tex].
3. Analyze the transformation [tex]\( -4 \)[/tex]: The expression [tex]\( f(x+2) - 4 \)[/tex] indicates a vertical shift. Subtracting 4 from the function [tex]\( f(x+2) \)[/tex] shifts the entire graph downward by 4 units.
- New shifted point [tex]\((x, f(x+2))\)[/tex] becomes [tex]\((x, f(x+2) - 4)\)[/tex].
- For example, the previous point [tex]\((-2, 0)\)[/tex] moves to [tex]\((-2, 0-4)\)[/tex], which is [tex]\((-2, -4)\)[/tex].
4. Combining the transformations: These transformations give us the final graph of [tex]\( g(x) = f(x+2) - 4 \)[/tex], which is the graph of the cubic root function [tex]\( \sqrt[3]{x} \)[/tex], shifted 2 units to the left and 4 units down.
So, the correct answer is: The graph of [tex]\( g(x) = f(x + 2) - 4 \)[/tex] is a cubic root function shifted 2 units to the left and 4 units down.
