poopey
Answered

Use a graphing calculator to approximate the vertex of the graph of the parabola defined by the following equation.

[tex]\[ y = x^2 + x + 6 \][/tex]

A. [tex]\((-0.5, 5.75)\)[/tex]
B. [tex]\((0.5, -5.75)\)[/tex]
C. [tex]\((-0.5, 6)\)[/tex]
D. [tex]\((0.5, 5.75)\)[/tex]

Please select the best answer from the choices provided:

A
B
C
D



Answer :

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