If I have a parabola $\mathcal{P}: y=ax^2+bx+c$, I know that $a$ provides me with the aperture of the parabola; $c$ is the known term. If it is zero the parabola passes through the origin otherwise not.
What is the role or the meaning that can be given to $b$ in the description of a parabola?
On
I think as you said in the comments it has a role on "shiftting" the parabola in the x-y plane since it partially determines the coordinates of the vertex. Also $b$ is related to the location of the zeros of the parabola so you know that if $b$ equals zero the parabola will be symmetrical respect the $y$-axis.
Hope it helps, have a good evening.
On
I look upon things differently.
If $~y = ax^2 + bx + c,~$
then $~\displaystyle \frac{y}{a} = x^2 + \frac{b}{a} ~x + \frac{c}{a} = \left( ~x + \frac{b}{2a} ~\right)^2 + \frac{4ac - b^2}{4a^2}.~$
Assigning variables:
you then have that
$$\frac{y}{a} = \left( ~x + M ~\right)^2 + D. \tag1 $$
Then, the graph of $~y~$ can be constructed from the graph of $~y = x^2,~$ by:
Shifting the graph $~M~$ units to the left.
Then shifting the graph $~D~$ units up.
Then stretching the graph by the factor $~a.$
If you notice the degree of the term involving $b$, you can deduce that there should be some connection to a line with equation $\mathcal{L}:y=bx+c$.
Specifically, $b$ is the slope of $\mathcal{L}$, which is always the tangent to $\mathcal{P}$ at the point $(0,c)$. We can see this algebraically by taking the derivative $\frac{d}{dx}(ax^2+bx+c)=2ax+b$ and plugging $x=0$ to obtain a constant slope $b$.
You can imagine that adjusting $b$ forces the parabola to move in a wave-like pattern to "match" its tangent line if that makes any sense.
Here is an interactive graph with both the tangent $\mathcal{L}$ and parabola $\mathcal{P}$ in view. Notice that adjusting $a$ does not affect the tangent.