The first stage is to factorise the equation, since #x^2# is negative I will multiply the equation by #-1# to make it easier to factorise.
Factorise:
#y = x^2 - 4x - 12#
#(x - 6)(x + 2)#
To obtain the #x#-intercept we need to make #y = 0#,
#0 = (x - 6)(x+2)#
We can now solve for the two values
#x_1 - 6 = 0#
#x_1 = 6#
#x_2 + 2 = 0#
#x_2 = -2#
To obtain the axis of symmetry for the parabola we add the two #x# values together then divide by #2#. This will give the #x# value for the vertex.
#x_v = (x_1 + x_2)/ 2= (6 - 2) / 2 = 2#
Now to get the #y# value for the vertex substitute #x = 2# into the original equation and solve:
#y_v = -x^2 + 4x +12#
#y_v = (2)^2 - 4(2) -12#
#y_v = 16#
Therefore the vertex is #(2,16)#
The final intercept we need is the #y#-intercept, this can be calculated by substituting #x = 0# into the original equation:
#y = -x^2 + 4x +12#
#y = -(0)^2 + 4(0) +12#
#y = 12#
#y#-intercept = #(0,12)#
