To find the [tex]\( y \)[/tex]-coordinate of the vertex for the quadratic function [tex]\( y = 2x^2 + 6 \)[/tex], we can follow these steps:
1. Understand the Form of the Quadratic Function:
The quadratic function is given in the form [tex]\( y = ax^2 + bx + c \)[/tex]. In this specific function, the coefficients are:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 0 \)[/tex]
- [tex]\( c = 6 \)[/tex]
2. Find the [tex]\( x \)[/tex]-coordinate of the Vertex:
For a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex], the [tex]\( x \)[/tex]-coordinate of the vertex can be found using the formula:
[tex]\[
x_{\text{vertex}} = -\frac{b}{2a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
x_{\text{vertex}} = -\frac{0}{2 \cdot 2} = 0
\][/tex]
3. Substitute the [tex]\( x \)[/tex]-coordinate into the Original Function:
To find the [tex]\( y \)[/tex]-coordinate of the vertex, substitute [tex]\( x_{\text{vertex}} \)[/tex] back into the original quadratic equation [tex]\( y = 2x^2 + 6 \)[/tex]:
[tex]\[
y_{\text{vertex}} = 2(0)^2 + 6 = 6
\][/tex]
Therefore, the [tex]\( y \)[/tex]-coordinate of the vertex for the quadratic function [tex]\( y = 2x^2 + 6 \)[/tex] is [tex]\(\boxed{6}\)[/tex].
