Calculate the vertex of the quadratic function y = 2x^2 + 4x - 6.

Answers:

• To find the vertex of the quadratic function ( y = 2x^2 + 4x - 6 ), we can use the vertex formula. The ( x )-coordinate of the vertex for a quadratic function in the standard form ( y = ax^2 + bx + c ) is given by:

  [ x = -\frac{b}{2a} ]

  In this function, the coefficients are:

  • ( a = 2 )
  • ( b = 4 )
  • ( c = -6 )

  Plugging in the values of ( a ) and ( b ):

  [ x = -\frac{4}{2 \cdot 2} = -\frac{4}{4} = -1 ]

  Now, we need to find the ( y )-coordinate of the vertex by substituting ( x = -1 ) back into the quadratic equation:

  [ y = 2(-1)^2 + 4(-1) - 6 ]

  Calculating this step by step:

  [ y = 2(1) - 4 - 6 ]

  [ y = 2 - 4 - 6 ]

  [ y = 2 - 10 = -8 ]

  Therefore, the vertex of the quadratic function ( y = 2x^2 + 4x - 6 ) is:

  [ \boxed{(-1, -8)} ]