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?

  • $\begingroup$ It seems like you answered your own question. $\endgroup$ – user2345215 Jan 24 '15 at 22:47
  • $\begingroup$ Just optionally do something. $\endgroup$ – ReliableMathBoy Jan 24 '15 at 22:48

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 $c$.

  • $\begingroup$ Yeah, your flat, regular, horizontal line. $\endgroup$ – ReliableMathBoy Jan 24 '15 at 22:44
  • $\begingroup$ Not necessarily horizontal (in which case slope equals zero.) It can be any line (depending on $b, c$. $\endgroup$ – Namaste Jan 24 '15 at 22:46
  • $\begingroup$ $b=0$ and $c=2$, so $y=2$, which makes a horizontal line go through $(0, 2)$. $\endgroup$ – ReliableMathBoy Jan 24 '15 at 22:46
  • $\begingroup$ Well, when $a$ gets closer to $0$, the parabola becomes flatter. $\endgroup$ – ReliableMathBoy Jan 24 '15 at 22:47
  • $\begingroup$ Indeed, it does. $\endgroup$ – Namaste Jan 24 '15 at 22:48

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.


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