Final Answer:
For the parabola defined by the equation y = -(x + 4)² - 7:
- Vertex: (-4, -7)
- Axis of symmetry: x = -4
- Focus: (-4, -8)
- Directrix: y = -6
- Direction of opening: Downward
- Length of the latus rectum: 4
Step-by-step explanation:
The vertex of the parabola is determined by the values within the parentheses of the equation. For y = -(x + 4)² - 7, the vertex is (-4, -7). The axis of symmetry, given by the vertical line that passes through the vertex, is x = -4.
To find the focus and directrix of the parabola, we need to consider its standard form 4p(y - k) = (x - h)², where (h, k) is the vertex and p is the distance from the vertex to the focus and directrix. In this case, h = -4 and k = -7. The negative coefficient in front of the squared term indicates a downward opening parabola. The focus is (-4, -7 - p), and the directrix is a horizontal line y = -7 + p. By comparing coefficients, we find p = 1, so the focus is (-4, -8), and the directrix is y = -6.
The length of the latus rectum, which is the focal width of the parabola, is equal to 4p. Substituting p = 1, we get a length of 4. This signifies the distance between the points on the parabola that intersect with a line parallel to the directrix and passing through the focus.