To find the vertex of the quadratic function [tex]\(f(x) = 2x^2 - 8x + 3\)[/tex], we can use the vertex formula for a quadratic function in the form [tex]\(ax^2 + bx + c\)[/tex], where the vertex [tex]\((h, k)\)[/tex] can be found using:
1. The x-coordinate of the vertex [tex]\(h\)[/tex]:
The x-coordinate of the vertex of a parabola described by [tex]\(ax^2 + bx + c\)[/tex] is given by the formula:
[tex]\[
h = \frac{-b}{2a}
\][/tex]
For the given function [tex]\(f(x) = 2x^2 - 8x + 3\)[/tex]:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -8\)[/tex]
Plugging in these values:
[tex]\[
h = \frac{-(-8)}{2 \cdot 2} = \frac{8}{4} = 2
\][/tex]
2. The y-coordinate of the vertex [tex]\(k\)[/tex]:
To find the y-coordinate of the vertex, substitute [tex]\(h\)[/tex] back into the original function:
[tex]\[
k = f(h) = f(2)
\][/tex]
Calculate [tex]\(f(2)\)[/tex]:
[tex]\[
f(2) = 2(2)^2 - 8(2) + 3 = 2 \cdot 4 - 16 + 3 = 8 - 16 + 3 = -5
\][/tex]
3. Vertex coordinates:
Therefore, the vertex of the function [tex]\(f(x) = 2x^2 - 8x + 3\)[/tex] is:
[tex]\[
(h, k) = (2, -5)
\][/tex]
Given the answers to choose from, the correct vertex is [tex]\((2, -5)\)[/tex].
Thus, the answer is [tex]\((2, -5)\)[/tex].
