Final answer:
The focus of the parabola y = 5x² is (0, 1/20) and the directrix is y = -1/20. The focal diameter can be calculated using the distance formula. To sketch the graph of the parabola, plot points by substituting x-values into the equation and connect them with a smooth curve.
Step-by-step explanation:
The given equation is in the form y = ax^2, which is the general equation of a parabola. The coefficient of x^2 determines the shape of the parabola. In this case, the coefficient is 5, which means that the parabola opens upward. To find the focus and directrix of the parabola, we can use the standard form of the equation: (x-h)^2 = 4p(y-k), where (h, k) is the vertex of the parabola. For the given equation, the vertex is (0,0) since there are no values added or subtracted from x or y. Thus, the focus and directrix are both on the line y = -p. In this case, p = 1/(4a) = 1/(4(5)) = 1/20. So, the directrix is y = -1/20 and the focus is (0, 1/20).
The focal diameter of a parabola is the distance between its focus and a point on the parabola that is perpendicular to the directrix. Since the directrix is a horizontal line, the focal diameter is the vertical distance between the focus and a point on the parabola. Using the distance formula, we can calculate the focal diameter. Let's take a point (x, y) on the parabola where y = 5x^2. So, the distance between the focus (0, 1/20) and the point (x, 5x^2) is given by: d = sqrt((x-0)^2 + (5x^2 - 1/20)^2). This is the general formula for the focal diameter of the parabola y = 5x^2.
To sketch the graph of the parabola, we can plot points that satisfy the given equation y = 5x^2. Choosing some x-values, we can calculate the corresponding y-values by substituting them into the equation. For example, if we choose x = 0, then y = 5(0)^2 = 0. Similarly, if we choose x = 1, then y = 5(1)^2 = 5. Plotting these points on a graph and connecting them with a smooth curve will give us the shape of the parabola.