1. **State the problem:** Find the vertex, y-intercept, and x-intercepts of the quadratic function $$y = 3x^2 - 12x + 9$$ using calculation. 2. **Vertex formula:** The vertex of a quadratic function $$y = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. 3. **Calculate the vertex:** $$a = 3, \quad b = -12, \quad c = 9$$ $$x = -\frac{-12}{2 \times 3} = \frac{12}{6} = 2$$ 4. **Find the y-coordinate of the vertex by substituting $$x=2$$ into the function:** $$y = 3(2)^2 - 12(2) + 9 = 3 \times 4 - 24 + 9 = 12 - 24 + 9 = -3$$ 5. **Vertex coordinates:** $$(2, -3)$$ 6. **Find the y-intercept:** The y-intercept occurs when $$x=0$$. $$y = 3(0)^2 - 12(0) + 9 = 9$$ 7. **Y-intercept coordinates:** $$(0, 9)$$ 8. **Find the x-intercepts:** Set $$y=0$$ and solve for $$x$$: $$0 = 3x^2 - 12x + 9$$ Divide both sides by 3: $$0 = \cancel{3}x^2 - \cancel{3}4x + \cancel{3}3 \Rightarrow 0 = x^2 - 4x + 3$$ 9. **Factor the quadratic:** $$x^2 - 4x + 3 = (x - 3)(x - 1)$$ 10. **Set each factor to zero:** $$x - 3 = 0 \Rightarrow x = 3$$ $$x - 1 = 0 \Rightarrow x = 1$$ 11. **X-intercept coordinates:** $$(3, 0)$$ and $$(1, 0)$$ **Final answers:** - Vertex: $$(2, -3)$$ - Y-intercept: $$(0, 9)$$ - X-intercepts: $$(3, 0)$$ and $$(1, 0)$$