Related to parabola

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.