1. **Problem (a):** Find the vertex, focus, and directrix of the parabola given by $$(y - 7)^2 = 8(x - 2)$$ 2. **Formula and rules:** This is a parabola that opens horizontally because it is in the form $$(y - k)^2 = 4p(x - h)$$ where $(h,k)$ is the vertex. 3. **Identify vertex:** From the equation, vertex is at $$(2, 7)$$. 4. **Find $p$:** Here, $4p = 8 \Rightarrow p = 2$. 5. **Focus:** Since the parabola opens right (positive $p$), focus is at $$(h + p, k) = (2 + 2, 7) = (4, 7)$$. 6. **Directrix:** The directrix is the vertical line $$x = h - p = 2 - 2 = 0$$. --- 1. **Problem (b):** Find the vertex and focus of the parabola given by $$y^2 - 4y = 12x - 2^2$$ 2. **Rewrite equation:** Note $2^2 = 4$, so rewrite as $$y^2 - 4y = 12x - 4$$. 3. **Complete the square for $y$:** $$y^2 - 4y + 4 = 12x - 4 + 4$$ $$ (y - 2)^2 = 12x$$ 4. **Rewrite in vertex form:** $$(y - 2)^2 = 12(x - 0)$$ 5. **Vertex:** $$(0, 2)$$. 6. **Find $p$:** $4p = 12 \Rightarrow p = 3$. 7. **Focus:** Since it opens right, focus is $$(0 + 3, 2) = (3, 2)$$. --- 1. **Problem (c):** Find vertex, focus, and directrix of $$(x - 6)^2 = 12(y - 2)$$ 2. **Form:** This is a vertical parabola in the form $$(x - h)^2 = 4p(y - k)$$. 3. **Vertex:** $$(6, 2)$$. 4. **Find $p$:** $4p = 12 \Rightarrow p = 3$. 5. **Focus:** Since $p > 0$, parabola opens upward, focus is $$(6, 2 + 3) = (6, 5)$$. 6. **Directrix:** Horizontal line $$y = k - p = 2 - 3 = -1$$. --- 1. **Problem (d):** Find vertex, focus, and directrix of $$x^2 + 16x = 4y - 28$$ 2. **Rewrite:** Move terms to isolate $y$: $$4y = x^2 + 16x + 28$$ $$y = \frac{1}{4}x^2 + 4x + 7$$ 3. **Complete the square for $x$:** $$x^2 + 16x = (x^2 + 16x + 64) - 64 = (x + 8)^2 - 64$$ 4. **Rewrite $y$:** $$y = \frac{1}{4}((x + 8)^2 - 64) + 7 = \frac{1}{4}(x + 8)^2 - 16 + 7 = \frac{1}{4}(x + 8)^2 - 9$$ 5. **Vertex:** $(-8, -9)$. 6. **Form:** $$(x + 8)^2 = 4p(y + 9)$$ multiply both sides by 4: $$4(y + 9) = (x + 8)^2$$ Rewrite as: $$(x + 8)^2 = 4p(y + 9)$$ with $4p = 4$ so $p = 1$. 7. **Focus:** Since $p > 0$, parabola opens upward, focus is at $$(h, k + p) = (-8, -9 + 1) = (-8, -8)$$. 8. **Directrix:** Horizontal line $$y = k - p = -9 - 1 = -10$$. **Final answers:** (a) vertex: $(2, 7)$, focus: $(4, 7)$, directrix: $x = 0$ (b) vertex: $(0, 2)$, focus: $(3, 2)$ (c) vertex: $(6, 2)$, focus: $(6, 5)$, directrix: $y = -1$ (d) vertex: $(-8, -9)$, focus: $(-8, -8)$, directrix: $y = -10$