The standard form of a parabolic equation that opens upwards or downwards is given as (x - h)² = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus of the parabola.
In this scenario, we have the vertex (h, k) as (1, -2) and the focus (a, b) as (1, -4). Since the y-coordinate of the focus is less than that of the vertex, we can establish that this parabola opens downwards.
The next step is to find the value of p. The distance, p, from the vertex to the focus is determined by the difference in the y-coordinate of the vertex and the focus. This gives us p = -2 - (-4), or p = 2.
With the values of h, k and p, the assigned equation becomes (x - 1)² = 4*2*(y + 2). Simplifying this equation puts it in standard form.
First, we expand (x - 1)²:
x² - 2x + 1 = 8*(y + 2)
Simplify the right-hand side gives:
x² - 2x + 1 = 8y + 16
Next, we re-arrange the equation to isolate terms with variables on one side and constants on the other:
x² - 2x - 8y + 1 - 16 = 0
When the constants are combined, the final standard form of the equation is:
x² - 2x - 8y - 15 = 0
So the standard form of the equation of the given parabola is x² - 2x - 8y -15 = 0.