This is a parabola, an inverted U-shaped curve. It doesn't intersect the x axis because there are no real zeroes. But when x=0, f(x)=-2 so the vertical axis is intercepted at -2. The graph can be written in the form f(x)=a(x-h)^2+k, where (h,k) is the vertex or origin. To find h and k we expand this: f(x)=ax^2-2axh+ah^2+k. From this a=-2, -2ah=-1, so h=-1/4 and ah^2+k=-2, so k=-2+1/8=-15/8 and f(x)=-2(x+1/4)^2-15/8. The vertical line of symmetry is x=-1/4 and the vertex (maximum) lies on this line at (-1/4,-15/8), below the x axis. These are properties of the parabola that help you to draw it. It helps to plot a few other points, for example, (1,-5) and (-1,-3) to get some idea of the spread of the arms of the inverted U.
Two other properties of the parabola may also help to graph it: directrix and focus. The directrix is a line defined by a, so that the line is outside the parabola at f(x)=1/(4a)=-1/8. The focus sits inside the parabola on the line of symmetry so that the vertex is midway between the focus and directrix. Since the vertex is at (-1/4,-15/8) and the directrix is f(x)=-1/8, the distance between the vertex and directrix is 15/8-1/8=14/8=7/4 and the focus must be at f(x)=-15/8-7/4=-29/8, making the focus (-1/4,-29/8). All points on the parabola lie equidistant from the directrix and focus. The directrix and focus control the spread of the parabolic arms.