Questions: Find the vertex of the function given below. y=2 x^2+4 x+1

Solution Steps

To find the vertex of a quadratic function in the form \( y = ax^2 + bx + c \), we can use the vertex formula. The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Once we have the x-coordinate, we substitute it back into the original equation to find the y-coordinate.

The given quadratic function is \( y = 2x^2 + 4x + 1 \). Here, the coefficients are \( a = 2 \), \( b = 4 \), and \( c = 1 \).

The x-coordinate of the vertex for a quadratic function \( y = ax^2 + bx + c \) is given by the formula: \[ x = -\frac{b}{2a} \] Substituting the values of \( a \) and \( b \): \[ x = -\frac{4}{2 \times 2} = -\frac{4}{4} = -1.0 \]

Substitute \( x = -1.0 \) back into the original equation to find the y-coordinate: \[ y = 2(-1.0)^2 + 4(-1.0) + 1 \] \[ y = 2(1.0) - 4.0 + 1 \] \[ y = 2.0 - 4.0 + 1 = -1.0 \]

Final Answer