Question

What is the the vertex of `y = (x+6)(x+4)`?

Answer 1

The vertex is the point `(x,y)=(-5,-1)`.

Explanation:

Let `f(x)=(x+6)(x+4)=x^{2}+10x+24`.

One approach is to just realize that the vertex occurs halfway between the `x`-intercepts of `x=-4` and `x=-6`. In other words, the vertex is at `x=-5`. Since `f(-5)=1*(-1)=-1`, this means the vertext is at `(x,y)=(-5,-1)`.

For a more general approach that works even when the quadratic function has no `x`-intercepts, use the method of Completing the Square :

`f(x)=x^[2}+10x+24=x^{2}+10x+(10/2)^{2}+24-25=(x+5)^{2}-1`.

This puts the quadratic function in "vertex form", which allows you to see that its minimum value of `-1` occurs at `x=-5`.

Here's the graph:

graph{(x+6)(x+4) [-20, 20, -10, 10]}