To find the vertex of the parabola given by the equation:
[tex]\[ y = -2x^2 + 4x + 1, \][/tex]
we use the standard formula for the coordinates of the vertex of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex].
1. Find the x-coordinate of the vertex:
The formula for the x-coordinate of the vertex, [tex]\( x \)[/tex], is:
[tex]\[ x = \frac{-b}{2a}, \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the coefficients from the quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex].
Here, [tex]\( a = -2 \)[/tex] and [tex]\( b = 4 \)[/tex]. Plugging these values into the formula gives:
[tex]\[ x = \frac{-4}{2(-2)} = \frac{-4}{-4} = 1. \][/tex]
2. Find the y-coordinate of the vertex:
Substitute the x-coordinate back into the original equation to find the y-coordinate:
[tex]\[ y = -2(1)^2 + 4(1) + 1 \][/tex]
[tex]\[ y = -2(1) + 4 + 1 \][/tex]
[tex]\[ y = -2 + 4 + 1 \][/tex]
[tex]\[ y = 3. \][/tex]
Therefore, the vertex of the parabola [tex]\( y = -2x^2 + 4x + 1 \)[/tex] is:
[tex]\[ (1.0, 3.0). \][/tex]
