Is it true that $a$ can't be zero in the quadratic function $y=ax^2+bx+c$?

Question

I read that for $y=ax^2+bx+c$ is a quadratic function where $a\neq0$, but is it true that $a$ really can't be zero? I think it is because if $a$ was zero, there wouldn't be a parabola. There would just be a flat line, so then it wouldn't be quadratic because the $x^2$-term indicates if the parabola opens upward or downward. Is this right or is it true about what I asked?

Answer 1

If $a=0$, you no longer have a parabola.

Instead, you have a line: $y = bx+c$, with slope equal to $b$, and a $y$-intercept at $c$.

Answer 2

I guess I'm right. If $a$ is zero, then $y=ax^2+bx+c$ would change to $y=bx+c$, leaving $b$ as the slope and $c$ as the y-intercept, leaving a flat line, not a parabola.