1. **Problem Statement:** Convert the given vertex forms of quadratic equations to their standard forms. 2. **Formula and Rules:** - Vertex form: $$y = a(x-h)^2 + k$$ where $(h,k)$ is the vertex. - Standard form: $$y = ax^2 + bx + c$$. - To convert, expand the squared binomial using FOIL and simplify. --- ### Problem 1: Convert $$y = 2(x + 3)^2$$ to standard form. 3. Expand the binomial: $$y = 2(x + 3)(x + 3)$$ 4. Apply FOIL: $$y = 2(x^2 + 3x + 3x + 9)$$ 5. Combine like terms inside parentheses: $$y = 2(x^2 + 6x + 9)$$ 6. Distribute 2: $$y = 2x^2 + 12x + 18$$ 7. **Final standard form:** $$y = 2x^2 + 12x + 18$$ --- ### Problem 2: Convert $$y = -(x + 1)^2 - 1$$ to standard form. 8. Expand the binomial: $$y = -(x + 1)(x + 1) - 1$$ 9. Apply FOIL: $$y = -(x^2 + x + x + 1) - 1$$ 10. Combine like terms inside parentheses: $$y = -(x^2 + 2x + 1) - 1$$ 11. Distribute the negative sign: $$y = -x^2 - 2x - 1 - 1$$ 12. Combine constants: $$y = -x^2 - 2x - 2$$ 13. **Final standard form:** $$y = -x^2 - 2x - 2$$ --- ### Problem 3: Convert $$y = -3(x - 1)^2 - 1$$ to standard form. 14. Expand the binomial: $$y = -3(x - 1)(x - 1) - 1$$ 15. Apply FOIL: $$y = -3(x^2 - x - x + 1) - 1$$ 16. Combine like terms inside parentheses: $$y = -3(x^2 - 2x + 1) - 1$$ 17. Distribute -3: $$y = -3x^2 + 6x - 3 - 1$$ 18. Combine constants: $$y = -3x^2 + 6x - 4$$ 19. **Final standard form:** $$y = -3x^2 + 6x - 4$$ --- **Summary:** - $$y = 2x^2 + 12x + 18$$ - $$y = -x^2 - 2x - 2$$ - $$y = -3x^2 + 6x - 4$$