What are the parameters of a parabola

Question

In the following figure I understand the $bx+c$ part. It is simply the equation of a line. But I don't understand where did $ax^2$ came from? What exactly is it? What does $a$ tell us about a parabola?

Answer 1

When you have $y=ax^2+bx+c$, the $a$ affects the shape of the parabola. An $a$ with large absolute value causes the parabola to be narrow and steep. An $a$ near zero causes the parabola to flatten out. Positive $a$ causes the parabola to open upward, while negative $a$ causes it to open downward.

In addition to what you noticed, the $bx$ causes the parabola to move around: positive $b$ cause the parabola to move down and left, while negative $b$ cause the parabola to move up and right. Finally, $c$ moves the parabola up (positive $c$) and down (negative $c$).

Answer 2

The blue line whose slope is $b$ and intercept is $c$ has $y=bx+c$ for its equation. The equation $y=ax^2+bx+c$ is describing the downward-pointing black parabola which lies below the blue line. Because it's downward pointing (and because it lies below the blue line), the coefficient $a$ must be negative, just as $b$ and $c$ are, according to the picture, both positive.