Question

For the parabola defined by the given equations, find the:

Vertex
Axis of symmetry
Focus
Directrix
Direction of opening
Length of the latus rectum
Given equation: y=−(x+4) ²−7

Answer 1

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.