Related to parabola The equation of a parabola is given by: 
$y= ax^2 + bx+c$
Why is it that when the coefficient of $x$ i.e. $a$ is positive we get an upward parabola and when it's negative we get a downward parabola? 
Also, I saw that increasing the value of $|a|$ narrows the parabola, why? 
Lastly, what is the role of $b$ in determining the structure of this parabola? 
 A: In the quadratic equation "$b$" and "$c$" terms are correlated to an axis translation, thus we can consider the simpler case $$y=ax^2$$ for which is clear the role of "$a$" to determine the sign of $y$.
To clarify the first point suppose to change the coordinates by translation by means of $y=(y+k)$ and $x=(x+h)$ then $$(y+k)=a(x+h)^2$$ $$y=ax^2+2hx+h^2-k$$ which is in the form $$y=ax^2+bx+c$$
I think this way is simpler because you don't need any calculus knowledge.
A: let $$f(x)=ax^2+bx+c$$ then we have $$f'(x)=2ax+b$$ and $$f''(x)=2a$$ if $$a>0$$ then we get a Minimum Point, if $a<0$ then we get a Maximum Point.
A: You're given $y = a x^2 + b x + c $
Start your analysis by completing the square in $x$, as follows:
$\begin{equation} 
\begin{split}
 y &= a ( x^2 + \dfrac{b}{a} x ) + c \\
&= a (x + \dfrac{b}{2a} )^2 - a (\dfrac{b}{2a})^2 + c  \\
&= a (x - x_0)^2 + y_0 \\
\end{split} 
\end{equation}$
So that,
$ (y - y_0) = a (x - x_0)^2 $
If $a \gt 0 $, then as $x $ gets away from $x_0$ in either direction (left or right),  $y$ will increase above $y_0$, and if we increase $a$ then $y$ will grow faster, thus for the same value of $|x - x_0|$ , $y $ will be higher, thus making the graph look narrower.  The same can be said when $a \lt 0$ where deviations of $x$ from $x_0$ cause $y$ to dip below $y_0$ , and as we make the negative $a$ more negative, the deviation of $y$ from $y_0$ will increase, again making the graph narrower.
