For the parabola $y = a x^2 + b x + c$ "a" is best thought of as determining the curvature (inverse to width)
so a>0 is a "smile", a=0 is a line, and a<0 is a "frown". The bigger |a| is, the narrower and steeper the parabola.

"b" determines the slope at x=0. If this is nonzero it pushes the vertex
(highest or lowest point) off to the side, and also up or down.

If "c" is zero then the y-intercept is at y=0 (and there is an x-intercept at x=0.) In general, the y-intercept is at y=c; so c slides the parabola up or down.

Other important quantities are $\large \frac{-b}{2a}$ (the x coordinate of the vertex and the line of symmetry), $c-\large \frac{b^2}{4a}$ (y coordinate of the vertex),
and $b^2-4ac$ (the discriminant, which is positive when there are two x intercepts, zero when there is one, and negative when there are none.

Good Hunting!
RD