Parabola Properties Ae48A4

1. **State the problem:** We are given the parabola equation $$(x - 2)^2 = 12(y + 4)$$ and need to find its vertex, focus, directrix, axis of symmetry, and length of the latus rectum. 2. **Identify the form:** The equation is in the form $$(x - h)^2 = 4a(y - k)$$ which represents a parabola opening upward or downward. 3. **Compare given equation to standard form:** Given: $$(x - 2)^2 = 12(y + 4)$$ Rewrite as $$(x - 2)^2 = 12(y - (-4))$$ So, $h = 2$, $k = -4$, and $4a = 12$. 4. **Calculate $a$:** $$4a = 12 \implies a = \frac{12}{4} = 3$$ 5. **Find the vertex:** The vertex is at $$(h, k) = (2, -4)$$ 6. **Determine the focus:** For the form $$(x - h)^2 = 4a(y - k)$$ with $a > 0$, the parabola opens upward. Focus is at $$(h, k + a) = (2, -4 + 3) = (2, -1)$$ 7. **Find the directrix:** Directrix is the line $$y = k - a = -4 - 3 = -7$$ 8. **Axis of symmetry:** Axis of symmetry is the vertical line $$x = h = 2$$ 9. **Length of latus rectum:** Length of latus rectum is $$4a = 12$$ **Final answers:** - Vertex: $$(2, -4)$$ - Focus: $$(2, -1)$$ - Directrix: $$y = -7$$ - Axis of symmetry: $$x = 2$$ - Length of latus rectum: $$12$$