什么是二次函数---**Extracted Content:** **Problem Description:** 5. As shown in the figure, the line y = -2x + 8 intersects the x-axis and y-axis at points B and C respectively. The parabola y = -x^2 + bx + c passes through B and C. Its vertex is M, and the axis of symmetry MN intersects the line BC at point N. **Diagram 1 Description:** * **Type:** Coordinate plane with a parabola and a line segment. * **Related Questions:** (1), (2). * **Coordinate Axes:** X-axis and Y-axis intersecting at origin O. Arrows indicate positive direction. * **Parabola:** Opens downwards, passing through points labeled A, O, C, and B. Vertex is labeled M. The axis of symmetry is a vertical dashed line passing through M, intersecting the x-axis at an unlabeled point and the line segment BC at point N. * **Line Segment:** BC is a line segment. * **Points:** A, O, B (on x-axis), C (on y-axis), M (vertex of parabola), N (intersection of axis of symmetry and BC), P (on BC), Q (on parabola). * **Other Elements:** A vertical line segment PD is drawn from point P to the x-axis at point D. Point Q is vertically above P, on the parabola. **Diagram 2 Description:** * **Type:** Coordinate plane with a parabola, lines, and points. * **Related Questions:** (3). * **Coordinate Axes:** X-axis and Y-axis intersecting at origin O. Arrows indicate positive direction. * **Parabola:** Opens downwards, passing through points labeled E, A, B, F. (Note: This is the same parabola as in Diagram 1). * **Lines:** Line segment BC is shown (from Diagram 1). A line segment EF is drawn, parallel to BC. A line passes through O and H. * **Points:** O (origin), E (on parabola), A (on parabola), G (on negative y-axis), B (on parabola/x-axis), D (on x-axis), H (on line y=-4x and line EF), F (on parabola). * **Other Elements:** Point G is on the negative y-axis. Line EF is parallel to BC, passing through G, and intersects the parabola at E and F. The line y = -4x intersects EF at H. **Sub-questions:** (1) Directly write the equation of the parabola. (2) As shown in Figure 1, point P is a moving point on line segment BC. Draw PD perpendicular to the x-axis through point P, intersecting the x-axis at point D and the parabola at point Q. Question: Does there exist a point P such that quadrilateral MNPQ is a rhombus? And explain the reason. (3) As shown in Figure 2, point G is a moving point on the negative half of the y-axis. Draw EF parallel to BC through point G, line EF intersects the parabola at points E, F. It intersects the line y=-4x at point H. If $\frac{1}{EG} - \frac{1}{FG} = \frac{1}{HG}$, find the coordinates of point G.