I'm assuming that by standard form, you mean that the parabola is expressed with the equation y=ax^2+bx+c. So, a is the quadratic coefficient.
First of all, it is a general rule that, given the plot of a function f(x), the plot of f(x)+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 -\frac{b}{2a}, 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 \pm\infty are both the same infinite, and the a coefficient tells us its positivity: if a is positive, we have that
\lim_{x\to\pm\infty} ax^2+bx+c = \infty
while if a is negative,
\lim_{x\to\pm\infty} ax^2+bx+c = -\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]}
