To find the vertex of the quadratic function [tex]\( f(x) = -2x^2 + 4x + 1 \)[/tex], we can follow these steps:
1. Identify the axis of symmetry: We are given that the axis of symmetry is [tex]\( x = 1 \)[/tex].
2. Compute the y-coordinate of the vertex: To find the y-coordinate of the vertex, substitute [tex]\( x = 1 \)[/tex] into the quadratic function.
[tex]\[
f(1) = -2(1)^2 + 4(1) + 1
\][/tex]
3. Evaluate the expression:
[tex]\[
-2(1)^2 + 4(1) + 1 = -2 \cdot 1 + 4 \cdot 1 + 1 = -2 + 4 + 1 = 3
\][/tex]
4. Combine the results into the coordinate format: The vertex of the function, therefore, is located at [tex]\( (1, 3) \)[/tex].
Thus, the vertex of the function [tex]\( f(x) = -2x^2 + 4x + 1 \)[/tex] is at the point [tex]\( (1, 3) \)[/tex].
So, the correct answer is [tex]\( (1, 3) \)[/tex].
