Find the vertex, focus and directrix of the parabola and sketc y^2 + 2y + 12x + 30 = 0

Question

please help me

Answer 1

The equation is

To change the expression [ y² + 2y ] into a perfect square trinomial add and subtract (half the x coefficient)²

 Here y coefficient = 2. so, (half the x coefficient)² = (2/2)²= 1.

The above equation is in standard form of the parabola.

The standard form of parabola equation is (y - k)^2 = 4p (x - h), where (h, k) = vertex and p = directed distance from vertex to focus.

p = directed distance from vertex to focus = -3.

Focus is p distance to the left of vertex on the axis of symmetry.

Focus = (h+p, k)

Answer 2

Make a table; choose some values for y and find the corresponding values for x.

y	x=-(y2 + 2y + 30)/12	(x, y )
- 3	-[(-3)2 + 2(-3)+30]/12 =-(9 - 6 + 30)/12 = -33/12=-2.75	(-2.75, -3)
- 2	-[(-2)2 + 2(-2)+30]/12 =-(4 - 4 + 30)/-12 = -30/12 = -2.5	(- 2.5, -2)
0	-[(0)2 + 2(0)+30]/12 =-(0 - 0 + 30)/12 = -30/12 = -2.5	(-2.5,0)
1	-[(1)2 + 2(1)+30]/12 =-(1 + 2+ 30)/12 = -33/12 = -2.75	(-2.75, 1)
2	-[(2)2 + 2(2)+30]/12 =-(4 + 4 + 30)/12 = -38/12 = -3.16667	(-3.16667,2)

Use these ordered pairs to graph the equation.

1.Draw a coordinate plane.

2.Plot the points.

3.Draw a smooth curve through these points.

Directrix is x = h - p = -2.41667-(-3) = -2.41667 + 3 = 0.5833.

Therefore vertex, focus and directrix are (-2.41667, -1) , (-5.41667, -1) and x = 0.5833 respectively.