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

1 Answer
Bill K.
Jan 4, 2016

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]}

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