Question

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

Answer 1

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`.