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Find the parabola defined by the given equations and determine its characteristics.

[y = -(x + 4)² - 7]

a) Vertex: (-4, -7), Axis of Symmetry: x = -4, Focus: N/A, Directrix: N/A, Direction of Opening: Downward, Length of Latus Rectum: N/A
b) Vertex: (-4, -7), Axis of Symmetry: y = -7, Focus: N/A, Directrix: N/A, Direction of Opening: Downward, Length of Latus Rectum: N/A
c) both a and b
d) None of the above

1 Answer

2 votes

Final answer:

The parabola given by y = -(x + 4)² - 7 has its vertex at (-4, -7), the axis of symmetry is x = -4, and it opens downward. Options stating that focus or directrix are N/A are incorrect as all parabolas have these features, even if not specified.

Step-by-step explanation:

The parabola defined by the equation y = -(x + 4)² - 7 does indeed have a vertex at (-4, -7). This can be seen by completing the square or recognizing the vertex form of a parabola, which is y = a(x - h)² + k where (h, k) is the vertex. In this case, x + 4 is the squared term and -7 is the constant, indicating the vertex coordinates directly.

The axis of symmetry is a vertical line that goes through the vertex, and since the x-coordinate of the vertex is -4, the equation of the axis of symmetry is x = -4. The direction of opening is determined by the coefficient of the squared term; because it is negative, the parabola opens downward.

Although the focus and directrix are not given, they are not "not applicable" (N/A) as they can be calculated for any parabola. Therefore, options stating that the focus or directrix are N/A are incorrect. The correct answer when it comes to the characteristics specified would be closest to option (a).

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