This function y = x² - 4x - 5 has been put into 'standard form' 
y =(x-2)² - 9.
For the function, what would be the equation of the line of symmetry 
for the graph of this function?

The line of symmetry is the line that bisects 
the graph of the equation, which is a parabola.
Here is the graph of the parabola of your 
problem, and the green line in its axis of 
symmetry:

 

Look at some of the points on that green line:

(2,0) is a point on that green line.
(2,2) is a point on that green line.
(2,-2) is a point on that green line.
(2,4) is a point on that green line.
(2,-6) is a point on that green line.
(2,8) is a point on that green line.
(2,-10) is a point on that green line.
(2,6) is a point on that green line.
(2,100) is a point on that line, or it would 
be if we drew the graph big enough.
(2,-100) is a point on that line, or it would 
be if we drew the graph big enough.
(2,100000000) is a point on that line, or it 
would be if we could draw the graph big 
enough.
(2, anything) is a point on that green line.

What do all those points on that green line 
have in common?  Answer: Every one of their 
x-coordinates is 2.  Therefore all you have 
to do to write the equation of that green 
line is to say what is true about every point 
on that line, and that is:

Every point has the x value of 2

     which is the same as saying

All x-values on that line are equal to 2.

     which is the same as saying

x is always equal to 2

     which is the same as saying 

x = 2

That's the equation of that green vertical line
that bisects the parabola of the graph of your
given equation. And that green line is the axis 
of symmetry of the parabola.  Its equation is
simply

        x = 2 

Edwin