Explanation:
first of all, a = 0 would turn the quadratic function and parabola curve
y = ax² + bx + c
into a straight line :
y = bx + c
now, philosophically, we could argue that a straight line is a special and extreme form of a parabola.
the smaller the absolute value of "a", the more outstretched the parabola becomes. or rather, the further out the vertex (the turn around point) moves.
the x- coordinate of the vertex is
x = -b/(2a)
as we can see, the limit of bringing a to 0 is bringing that x- value to infinity. but still there is a vertex.
and that is the difference.
a line does not have a "turn-around" point (where the slope = the rate of change is changing its sign). not in finite nor in infinite spaces. the slope (or rate of change) is constant across all points of the line, even as limit for x going to infinity. the line is its own tangent at all points with one constant slope, for which 0 is just one special case for horizontal lines.
but a quadratic equation has an ever-changing slope : each point on the curve has its own individual tangent with its own individual slope (rate of change).
with the limit of that slope for "a" going to 0 and "x" going to infinity (making "ax²" an 0×infinity term) being 0 for every parabola (as the tangent at the vertex is a horizontal line with the slope = 0). not just special cases.
in other words : a line is a tangent to a quadratic function, but it is not a quadratic function itself.