Question 609154
<pre>

I do it the "vertex formula" way

y = 2x² - 8x - 1

Compare your equation to this model:

y = ax² + bx + c

It's easy to see that a=2, b=-8, c = -1

We know that it opens upward because "a" is a positive number.

The vertex formula is:

The vertex or turning point is (h,k) where h = {{{-b/(2a)}}} and 
k = ah²+bh+c  and the axis of symmetry is the vertical line whose 
equation is x = h

So for your vertex, h = {{{-(-8)/((2)(2))}}} = {{{8/4}}} = 2

and k = ah²+bh+c = (2)(2)²+(-8)(2)+(-1) = 2(4)-16-1 = 8-16-1 = -9 

So the vertex is (h,k) = (2,-9) 

The y-intercept is found by letting x=0 in the original equation:
       
    y = 2x² - 8x - 1
    y = 2(0)² - 8(0) - 1
    y = -1

The y-intercept is the point (0,-1)

So we plot the vertex the axis of symmetry and the y-intercept:

{{{drawing(225,400,-2,7,-11,5, graph(225,400,-2,7,-11,5),

circle(2,-9,.1), circle(0,-1,.1), green(line(2,10,2,-20)) )}}}

Now you can draw in the parabola so that it is symmetrical with
the green line of symmetry.

 {{{drawing(225,400,-2,7,-11,5, graph(225,400,-2,7,-11,5, 2x^2-8x-1),

circle(2,-9,.1), circle(0,-1,.1), green(line(2,10,2,-20)) )}}}

All x-values are in the domain, so the domain is all reals ({{{-infinity}}}, {{{infinity}}})

The only y-values are those from the vertex which is -9 upward, 
so the range is (-9,{{{infinity}}}).

Edwin</pre>