Question

What are the characteristics of the parabola defined by the equation y = -(x + 4)^2 - 7?

a) Vertex:
b) Axis of symmetry:
c) Focus:
d) Directrix:
e) Direction of opening:
f) Length of the latus rectum:

Answer 1

The parabola defined by y = -(x + 4)^2 - 7 has a vertex at (-4, -7), axis of symmetry x = -4, focus (-4, -7.25), directrix y = -6.75, it opens downward, and the length of the latus rectum is 1.

Step-by-step explanation:

The characteristics of the parabola defined by the equation y = -(x + 4)^2 - 7 are as follows:

• Vertex: The vertex form of a parabola is given by y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. For the given equation, the vertex is at (-4, -7).
• Axis of symmetry: The axis of symmetry is a vertical line that goes through the vertex. For this parabola, the axis of symmetry is x = -4.
• Focus: To find the focus, use the formula 4p = 1/a, where p is the distance from the vertex to the focus. Since a = -1 in the given equation, p = -1/4. The focus is at (-4, -7 - 1/4) or (-4, -7.25).
• Directrix: The directrix is a horizontal line at the distance p from the vertex on the opposite side of the focus. So, it is y = -7 + 1/4 or y = -6.75.
• Direction of opening: Since the coefficient a is negative, the parabola opens downward.
• Length of the latus rectum: The length of the latus rectum is given by the absolute value of 4p. Here, it is |4*(-1/4)|, which is 1.