To find the coordinates of the vertex of the given quadratic equation [tex]\( y = -2x^2 - 14 \)[/tex], we can proceed as follows:
### 1. Understanding the Standard Form
The equation [tex]\( y = -2x^2 - 14 \)[/tex] is already in the standard form of a quadratic equation, [tex]\( y = ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = 0 \)[/tex]
- [tex]\( c = -14 \)[/tex]
### 2. Finding the x-coordinate of the Vertex
The x-coordinate of the vertex for a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Given [tex]\( b = 0 \)[/tex] and [tex]\( a = -2 \)[/tex], substitute these values into the formula:
[tex]\[ x = -\frac{0}{2 \cdot (-2)} \][/tex]
[tex]\[ x = 0 \][/tex]
Therefore, the x-coordinate of the vertex is [tex]\( x = 0 \)[/tex].
### 3. Finding the y-coordinate of the Vertex
To find the y-coordinate of the vertex, substitute [tex]\( x = 0 \)[/tex] back into the original equation [tex]\( y = -2x^2 - 14 \)[/tex]:
[tex]\[ y = -2(0)^2 - 14 \][/tex]
[tex]\[ y = -2 \cdot 0 - 14 \][/tex]
[tex]\[ y = -14 \][/tex]
Therefore, the y-coordinate of the vertex is [tex]\( y = -14 \)[/tex].
### 4. Vertex Coordinates
Combining the x-coordinate and y-coordinate, the coordinates of the vertex are:
[tex]\[ (x, y) = (0, -14) \][/tex]
Thus, the coordinates of the vertex of the parabola [tex]\( y = -2x^2 - 14 \)[/tex] are:
[tex]\[ \boxed{(0, -14)} \][/tex]
