Question 596480
The axis of symmetry is a vertical line through
the x-coordinate of the vertex
The vertex is at {{{ -b/(2a) }}} when the equation 
is in the form {{{ ax^2 + b*x + c }}}
{{{ -b/(2a) = -1/(2*(-2)) }}}
{{{ -b/(2a) = 1/4 }}}
Now find {{{ y( 1/4 ) }}}
{{{ y(1/4) = -2*(1/4)^2 + 1/4 - 4 }}}
{{{ y(1/4) = -1/8 + 2/8 - 32/8 }}}
{{{ y(1/4) = -31/8 }}}
The vertex is at ( 1/4, -31/8 )
The axis of symmetry is {{{ x = 1/4 }}}
-------------------------------
Find the x-intercepts by setting {{{ y = 0 }}}
:{{{ y = -2x^2 + x - 4 }}} 
:{{{ -2x^2 + x - 4 = 0 }}} 
Use the quadratic formula
{{{ x = (-b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = -2 }}}
{{{ b = 1 }}}
{{{ c = -4 }}}
{{{ x = (-1 +- sqrt( 1^2 - 4*(-2)*(-4) )) / (2*(-2)) }}}
{{{ x = (-1 +- sqrt( 1 - 32 )) / (-4) }}}
{{{ x = (1 + sqrt( 31 ) *i) / 4 }}}
and
{{{ x = (1 - sqrt( 31 ) *i) / 4 }}}
The roots are imaginary, so there are no x-intercepts
here's a plot:
{{{ graph( 400, 400, -4, 4, -10, 2, -2x^2 + x - 4 ) }}}