To find the vertex of the parabola given by the equation [tex]\( y = -x^2 - 2x - 5 \)[/tex], we will follow these steps:
1. Identify the coefficients: The general form of a quadratic equation is [tex]\( y = ax^2 + bx + c \)[/tex]. From the given equation [tex]\( y = -x^2 - 2x - 5 \)[/tex], we can identify the coefficients as:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = -5 \)[/tex]
2. Calculate the x-coordinate of the vertex: The x-coordinate of the vertex of a parabola can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex]. Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula, we get:
[tex]\[
x = -\frac{-2}{2 \cdot -1} = \frac{2}{-2} = -1
\][/tex]
3. Calculate the y-coordinate of the vertex: Once we have the x-coordinate, we can find the y-coordinate by substituting [tex]\( x = -1 \)[/tex] back into the original equation [tex]\( y = -x^2 - 2x - 5 \)[/tex]. So, we have:
[tex]\[
y = -(-1)^2 - 2(-1) - 5 = -1 + 2 - 5 = -4
\][/tex]
4. Conclusion: The vertex of the parabola [tex]\( y = -x^2 - 2x - 5 \)[/tex] is at the point [tex]\((-1, -4)\)[/tex].
Therefore, the vertex is [tex]\((-1, -4)\)[/tex].
