To find the vertex of the quadratic function given by [tex]\( y = 2x^2 + 4x + 1 \)[/tex], we can use the vertex formula. The general form for a quadratic equation is [tex]\( y = ax^2 + bx + c \)[/tex].
The x-coordinate of the vertex can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = 2 \)[/tex] and [tex]\( b = 4 \)[/tex].
1. Calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{4}{2 \cdot 2} = -\frac{4}{4} = -1 \][/tex]
2. Substitute [tex]\( x = -1 \)[/tex] back into the original quadratic 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]\[ y = -1 \][/tex]
Thus, the coordinates of the vertex of the given quadratic function are [tex]\( (-1, -1) \)[/tex].
The correct answer is:
B. [tex]\( (-1, -1) \)[/tex]
