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]}
