Final answer:
To complete the square for the given parabolic equation y = -x^2 - 4x - 9, we can rewrite the equation in the form y = a(x - h)^2 + k, find the vertex, axis of symmetry, focus, directrix, direction of opening, and latus rectum.
Step-by-step explanation:
To complete the square, we want to rewrite the given equation in the form y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Start by rearranging the equation as follows:
y = -(x^2 + 4x) - 9
Next, complete the square by adding and subtracting the square of half the coefficient of x:
y = -(x^2 + 4x + (4/2)^2) - 9 - (4/2)^2
Simplify and rewrite the equation:
y = -(x + 2)^2 - 13
From this, we can identify the values of the vertex (h, k), which is (-2, -13).
The axis of symmetry is a vertical line passing through the vertex, so the equation of the axis of symmetry is x = -2.
Since the coefficient of x^2 is negative, the parabola opens downwards.
The focus and directrix can be calculated using the formula:
p = 1/(4a)
For this equation, a = -1, so p = -1/4.
The focus is located at (h, k + p), which gives us (-2, -13 - 1/4).
The directrix is a horizontal line given by the equation y = k - p, so y = -13 + 1/4.
The latus rectum of a parabola is the line segment perpendicular to the axis of symmetry and passing through the focus. Its length is given by 4|a|, so the latus rectum for this parabola is 4.