To find the coordinates of the vertex of the given parabola [tex]\( y = 2x^2 - 8 \)[/tex] algebraically, we will follow these steps:
1. Understand the General Form: The given quadratic function is of the form [tex]\( y = ax^2 + bx + c \)[/tex]. In this case, [tex]\( a = 2 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = -8 \)[/tex].
2. Find the x-coordinate of the Vertex: For any quadratic function in the form [tex]\( y = ax^2 + bx + c \)[/tex], the x-coordinate of the vertex, [tex]\( x = h \)[/tex], can be found using the formula:
[tex]\[
h = \frac{-b}{2a}
\][/tex]
Substitute [tex]\( a = 2 \)[/tex] and [tex]\( b = 0 \)[/tex] into the formula:
[tex]\[
h = \frac{-0}{2 \times 2} = 0
\][/tex]
3. Find the y-coordinate of the Vertex: To find the corresponding y-coordinate, substitute [tex]\( x = h \)[/tex] back into the original equation:
[tex]\[
y = 2(0)^2 - 8
\][/tex]
Simplify the equation:
[tex]\[
y = 0 - 8 = -8
\][/tex]
4. Write Down the Vertex Coordinates: The vertex of the parabola is the point [tex]\((h, k)\)[/tex]. From our calculations, we have [tex]\( h = 0 \)[/tex] and [tex]\( k = -8 \)[/tex].
Therefore, the coordinates of the vertex of the parabola [tex]\( y = 2x^2 - 8 \)[/tex] are [tex]\((0.0, -8.0)\)[/tex].
