1. **State the problem:** We need to graph the transformed cube root function given by $$y = -2 \sqrt[3]{x - 4} + 1.$$ 2. **Identify the base function and transformations:** The base function is $$y = \sqrt[3]{x}$$ which is the cube root function. - The transformation $$x - 4$$ inside the cube root shifts the graph right by 4 units. - The coefficient $$-2$$ in front of the cube root reflects the graph across the x-axis and vertically stretches it by a factor of 2. - The $$+1$$ outside the cube root moves the graph up by 1 unit. 3. **Translate key points:** The basic cube root points are: $$ (0, 0), (1, 1), (-1, -1). $$ Apply the transformations to each point: - For $$x=0$$: $$y = -2 \sqrt[3]{0 - 4} +1 = -2 \sqrt[3]{-4} + 1 = -2(-\sqrt[3]{4}) + 1 = 2\sqrt[3]{4} + 1.$$ - For $$x=5$$ (since $$5-4=1$$): $$y = -2 \sqrt[3]{1} + 1 = -2 (1) + 1 = -1.$$ - For $$x=3$$ (since $$3-4=-1$$): $$y = -2 \sqrt[3]{-1} +1 = -2(-1) +1 = 3.$$ 4. **Plot these points and sketch:** The graph is vertically stretched and flipped, shifted 4 to the right, and up 1. It passes through approximately points: $$\left(0, 2\sqrt[3]{4}+1\right), (3, 3), (5, -1).$$ 5. **Final answer:** The transformed cube root function $$y = -2 \sqrt[3]{x - 4} + 1$$ is the cube root graph shifted right by 4, reflected and stretched vertically by 2, and shifted up by 1.