To determine whether the quadratic function [tex]\( g(x) = x^2 + 6x + 11 \)[/tex] has a minimum or maximum vertex, let's proceed step-by-step.
### Step 1: Identify the Coefficients
A general quadratic function is given by [tex]\( ax^2 + bx + c \)[/tex]. For the function [tex]\( g(x) = x^2 + 6x + 11 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- [tex]\( c = 11 \)[/tex]
### Step 2: Determine the Direction of the Parabola
For a quadratic function [tex]\( ax^2 + bx + c \)[/tex]:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards and the vertex represents a minimum point.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards and the vertex represents a maximum point.
In this case, [tex]\( a = 1 \)[/tex] which is greater than 0, so the parabola opens upwards. Thus, the vertex represents a minimum point.
### Step 3: Find the Vertex
The x-coordinate of the vertex for the quadratic function [tex]\( ax^2 + bx + c \)[/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]\[ x = -\frac{6}{2 \cdot 1} = -\frac{6}{2} = -3 \][/tex]
Now, to find the y-coordinate, we substitute [tex]\( x = -3 \)[/tex] back into the original quadratic equation [tex]\( g(x) \)[/tex]:
[tex]\[ g(-3) = (-3)^2 + 6(-3) + 11 \][/tex]
[tex]\[ g(-3) = 9 - 18 + 11 \][/tex]
[tex]\[ g(-3) = 2 \][/tex]
Thus, the vertex of the function [tex]\( g(x) = x^2 + 6x + 11 \)[/tex] is [tex]\((-3, 2)\)[/tex].
### Summary
For the quadratic function [tex]\( g(x) = x^2 + 6x + 11 \)[/tex]:
- The vertex is at [tex]\((-3, 2)\)[/tex], which is a minimum point since the parabola opens upwards.
