Cubic Shift 2Ee45E

1. **State the problem:** We need to graph the function $$y = (x - 4)^3 + 4$$ by transforming the parent cubic function $$y = x^3$$. 2. **Recall the parent function and transformations:** The parent function is $$y = x^3$$, which is a cubic curve increasing from bottom-left to top-right. The transformation $$y = (x - h)^3 + k$$ shifts the graph of $$y = x^3$$ horizontally by $$h$$ units and vertically by $$k$$ units. 3. **Identify the transformations:** - Horizontal shift: $$h = 4$$ units to the right (because of $$x - 4$$). - Vertical shift: $$k = 4$$ units up. 4. **Apply the transformations to key points:** The parent function has key points: - $$(-2, -8)$$ - $$(-1, -1)$$ - $$(0, 0)$$ - $$(1, 1)$$ - $$(2, 8)$$ Shift each point right by 4 and up by 4: - $$(-2 + 4, -8 + 4) = (2, -4)$$ - $$(-1 + 4, -1 + 4) = (3, 3)$$ - $$(0 + 4, 0 + 4) = (4, 4)$$ - $$(1 + 4, 1 + 4) = (5, 5)$$ - $$(2 + 4, 8 + 4) = (6, 12)$$ 5. **Plot the transformed points and sketch the curve:** The graph is the cubic curve passing through these transformed points, maintaining the shape of the parent cubic but shifted accordingly. **Final answer:** The graph of $$y = (x - 4)^3 + 4$$ is the graph of $$y = x^3$$ shifted 4 units right and 4 units up, passing through points $$(2, -4), (3, 3), (4, 4), (5, 5), (6, 12)$$.