I am assuming the function is #y=2x^2 + 4x-3#
We need get this into the form:
#a( x - h)^2 + k#
Where #a# is the coefficient of #x^2#, #h# is the axis of symmetry and #k# is the maximum/minimum value of the function.
Start by placing a bracket around the terms containing the variable:
#(2x^2 + 4x) -3#
Factor out the coefficient of #x^2#
#2(x^2 + 2x)-3#
Add the square of half the coefficient of #x# inside the bracket and subtract it outside the bracket:
Note: We have to multiply the square of half the coefficient by #a# when subtracting it outside the bracket, this is because we factored this value out at the beginning. This catches a lot of people out.
#2(x^2 + 2x + (1)^2)-3 -2(1)^2#
Make: # (x^2 + 2x + (1)^2)# into the square of a binomial:
#(x +1)^2#
So far we have:
#2(x+1)^2-3-2(1)^2=> 2(x+1)^2-5#
So: #h= -1# , #k=-5#
And coordinates of vertex are:
#( -1 , -5 )#
graph{y=2x^2 + 4x-3 [-10, 10, -10, 10]}
