Answer :
To find the vertex of the parabola given by the equation [tex]\( y = -x^2 - 2x + 6 \)[/tex], we can follow these steps:
1. Identify the coefficients: The equation is in the standard quadratic form [tex]\( y = ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = 6 \)[/tex]
2. Calculate the x-coordinate of the vertex: The x-coordinate of the vertex of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{-2}{2(-1)} = \frac{2}{-2} = -1 \][/tex]
3. Calculate the y-coordinate of the vertex: To find the y-coordinate, substitute the x-coordinate back into the original equation:
[tex]\[ y = -(-1)^2 - 2(-1) + 6 \][/tex]
Simplify the expression:
[tex]\[ y = -1 + 2 + 6 = 7 \][/tex]
Thus, the vertex of the parabola [tex]\( y = -x^2 - 2x + 6 \)[/tex] is [tex]\((-1, 7)\)[/tex].
1. Identify the coefficients: The equation is in the standard quadratic form [tex]\( y = ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = 6 \)[/tex]
2. Calculate the x-coordinate of the vertex: The x-coordinate of the vertex of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{-2}{2(-1)} = \frac{2}{-2} = -1 \][/tex]
3. Calculate the y-coordinate of the vertex: To find the y-coordinate, substitute the x-coordinate back into the original equation:
[tex]\[ y = -(-1)^2 - 2(-1) + 6 \][/tex]
Simplify the expression:
[tex]\[ y = -1 + 2 + 6 = 7 \][/tex]
Thus, the vertex of the parabola [tex]\( y = -x^2 - 2x + 6 \)[/tex] is [tex]\((-1, 7)\)[/tex].
