To determine the vertex of the parabola defined by the equation [tex]\( y = x^2 + x + 6 \)[/tex], we need to use the vertex formula for a quadratic equation of the form [tex]\( y = ax^2 + bx + c \)[/tex]. In this case, the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = 6 \)[/tex]
The vertex of a parabola given by the equation [tex]\( y = ax^2 + bx + c \)[/tex] has the coordinates:
- The x-coordinate of the vertex is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
First, we calculate the x-coordinate of the vertex:
[tex]\[
x = -\frac{b}{2a} = -\frac{1}{2 \times 1} = -\frac{1}{2} = -0.5
\][/tex]
Next, we find the y-coordinate of the vertex by substituting [tex]\( x = -0.5 \)[/tex] back into the original equation [tex]\( y = x^2 + x + 6 \)[/tex]:
[tex]\[
y = (-0.5)^2 + (-0.5) + 6
\][/tex]
[tex]\[
y = 0.25 - 0.5 + 6
\][/tex]
[tex]\[
y = 5.75
\][/tex]
Therefore, the vertex of the parabola is at [tex]\( (-0.5, 5.75) \)[/tex].
Among the given options, the coordinates [tex]\((-0.5, 5.75)\)[/tex] correspond to option (b).
Answer: B
