# In the standard form of the equation for a parabola, what is represented by a?

Jan 21, 2015

I'm assuming that by standard form, you mean that the parabola is expressed with the equation $y = a {x}^{2} + b x + c$. So, $a$ is the quadratic coefficient.

First of all, it is a general rule that, given the plot of a function $f \left(x\right)$, the plot of $f \left(x\right) + c$ will simply be translated vertically, upward if $c$ is positive, or downward if $c$ is negative. So this is the role of the coefficient $c$.

As for the coefficient $a$ and $b$, they give us information about the shape of the parabola and its position on the plane.

In fact, we know that the abscissa of the vertix is given by $- \setminus \frac{b}{2 a}$, so both $a$ and $b$ coefficients tell us where the vertix is.

Moreover, we know that the behaviour at infinity of a polynomial is given by the highest power of $x$; and parabola is nothing but a polynomial of degree 2. So, the limit at both $\setminus \pm \setminus \infty$ are both the same infinite, and the $a$ coefficient tells us its positivity: if $a$ is positive, we have that
$\setminus {\lim}_{x \setminus \to \setminus \pm \setminus \infty} a {x}^{2} + b x + c = \setminus \infty$
while if $a$ is negative,
$\setminus {\lim}_{x \setminus \to \setminus \pm \setminus \infty} a {x}^{2} + b x + c = - \setminus \infty$

As an example, you can confront the plot of the simplest parabolas with $a > 0$ or $a < 0$, i.e. $y = {x}^{2}$ ($a = 1$) and $y = - {x}^{2}$ ($a = - 1$). Here's their graphs: you can see that the first one faces upwards, the second downward.

$y = {x}^{2}$
graph{x^2 [-4.77, 5.23, -0.897, 4.313]}

$y = - {x}^{2}$
graph{-x^2 [-5.02, 4.98, -4.94, 0.27]}