1. **Problem Statement:** Find the function and analyze the expression $y = (x + 1)^3 - 1$. 2. **Formula and Rules:** This is a cubic function shifted horizontally and vertically. The general cubic function is $y = x^3$. 3. **Step-by-step Solution:** - The term $(x + 1)^3$ means the cubic function is shifted 1 unit to the left. - The $-1$ outside the cube shifts the graph 1 unit down. 4. **Intermediate Work:** - Expand $(x + 1)^3 = x^3 + 3x^2 + 3x + 1$ - Substitute back: $y = x^3 + 3x^2 + 3x + 1 - 1 = x^3 + 3x^2 + 3x$ 5. **Explanation:** - The function $y = (x + 1)^3 - 1$ is a cubic curve shifted left by 1 and down by 1. - The graph passes through the point $(-1, -1)$, which is the new origin of the cubic curve. **Final answer:** $$y = (x + 1)^3 - 1$$