1. **State the problem:** We need to sketch the graph of the function $$y = 2(x - 2)^3 + 1$$ and understand its behavior. 2. **Identify the base function:** The base function is $$y = x^3$$, which is a cubic curve passing through the origin with an inflection point at (0,0). 3. **Apply transformations:** - The term $$(x - 2)$$ shifts the graph 2 units to the right. - The coefficient 2 stretches the graph vertically by a factor of 2, making it steeper. - The +1 shifts the graph 1 unit upwards. 4. **Find key points:** - At $$x=2$$, $$y = 2(2-2)^3 + 1 = 2(0)^3 + 1 = 1$$, so the inflection point is at $$(2,1)$$. - At $$x=3$$, $$y = 2(3-2)^3 + 1 = 2(1)^3 + 1 = 3$$. - At $$x=1$$, $$y = 2(1-2)^3 + 1 = 2(-1)^3 + 1 = 2(-1) + 1 = -1$$. 5. **Plot behavior:** - For $$x > 2$$, the function increases steeply because of the cubic term and vertical stretch. - For $$x < 2$$, the function decreases steeply. 6. **Sketch the graph:** - Start at the inflection point $$(2,1)$$. - Plot points like $$(1,-1)$$ and $$(3,3)$$. - Draw a smooth curve passing through these points, steeply increasing to the right and decreasing to the left. **Final answer:** The graph of $$y = 2(x - 2)^3 + 1$$ is a vertically stretched cubic curve shifted right by 2 and up by 1, with an inflection point at $$(2,1)$$.