determine vertex, focus and directrix of parabola. then graph the parabola.

second equation:
y^2 -4y -12x= 8

To determine the vertex, focus, and directrix of a parabola, we need to rewrite the equation in a standard form. The standard form of a parabolic equation is either of the following:

1. (x - h)^2 = 4p(y - k) (for a parabola opening upwards or downwards)
2. (y - k)^2 = 4p(x - h) (for a parabola opening rightwards or leftwards)

In the given equation: y^2 - 4y - 12x = 8, we need to convert it to one of the standard forms.

First, let's move the constant term to the other side of the equation to isolate the variables:

y^2 - 4y = 12x + 8

Next, complete the square for the variable terms. Take half of the coefficient of the y-term and square it, then add it to both sides of the equation:

y^2 - 4y + (4/2)^2 = 12x + 8 + (4/2)^2
y^2 - 4y + 4 = 12x + 8 + 4

Simplifying further, we have:

(y - 2)^2 = 12x + 12

Now the equation is in a standard form for a parabola opening rightwards or leftwards. We can compare it to the form (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p is the distance between the vertex and focus/directrix.

Comparing the equation, we have:

(h, k) = (0, 2)
4p = 12

From the equation, we can determine that the vertex is at (0, 2).

To find the focus and directrix, we need to calculate the value of p. We know that p = 3, as 4p = 12.

For a parabola opening rightwards or leftwards, the focus is located at (h + p, k), and the directrix has an equation of x = h - p.

Focus: (0 + 3, 2) = (3, 2)
Directrix: x = 0 - 3 = -3

Now we have determined the vertex, focus, and directrix of the parabola from the given equation.

To graph the parabola, we can plot the vertex, focus, and directrix on a coordinate plane, and then draw the curve symmetrically based on these points.