Final answer:
The focus of the parabola is the vertex point (0, 0), the directrix is y = -5/3, and the focal diameter is 5/3. To sketch the graph, plot the vertex at (0, 0) and use the shape of the parabola to plot additional points.
Step-by-step explanation:
The given equation is 5x + 3y² = 0. This equation represents a parabola. To find the focus, directrix, and focal diameter of the parabola, we can rewrite the equation in vertex form: y = a(x - h)² + k, where (h, k) is the vertex of the parabola and 'a' determines the shape of the parabola. In this case, we can rewrite the equation as y = (-5/3)(x - 0)² + 0.
The vertex of the parabola is (0, 0), which represents the focus. Since the coefficient of the squared term is negative, the parabola opens downwards. The directrix is the line y = k - a, which in this case is y = -5/3. The focal diameter is the distance between the focus and the directrix, which is equal to the absolute value of 'a', so the focal diameter is 5/3.
To sketch the graph of the parabola, start by plotting the vertex at (0, 0). Then, use the shape of the parabola determined by 'a' to plot additional points on both sides of the vertex. Finally, connect these points to form a smooth curve.