How do you find the axis of symmetry and vertex point of the function: # y=x^2+8x+12#?

1 Answer
Oct 1, 2015

Use the Method of Completing the Square to find that the axis of symmetry is #x=-4# and the vertex is at #(x,y)=(-4,-4)#. Details to follow.

Explanation:

Complete the square:

#y=f(x)=x^2+8x+12=(x^2+8x+16)+(12-16)=(x+4)^2-4#

Since #(x+4)^2# is minimized at #x=-4#, the vertex (low point) for this upward-pointing parabola is at #(x,y)=(-4,f(-4))=(-4,16-32+12)=(-4,-4)#.

The vertical line at #x=-4# is also the axis of symmetry of this parabola. For any given #c>0#, the values of #f(-4+c)# and #f(-4-c)# are equal (this is the algebraic form of the symmetry about the vertical line #x=-4#). We can check this: #f(-4+c)=(-4+c+4)^2-4=c^2-4# and #f(-4-c)=(-4-c+4)^2-4=(-c)^2-4=c^2-4#.