Can someone explain the parabola standard form?

a  =   the width of the parabola

If  l a l < 1...the graph is wider than the graph y = x^2, if  l a l > 1....the graph is narrower than the graph y =  x^2  and if a < 0....we have an "upside" down parabola

c  =  the y intercept

b...when this isn't 0,  it shifts the vertex of the graph off of the y axis....either left or right

Look at these graphs to get a feel for this :

https://www.desmos.com/calculator/shyvwfty0k

Yes...without  "b,"  all vertcal parabolas would have a vertex on the y axis....but, of course, the vertex can definitely be away from the y axis

what does b  represent?

\(\begin{array}{|lrcll|} \hline &y &=& ax^2+bx+c \\ &y' &=& 2ax+b \\ x=0: &y' &=& b \\ \hline \end{array} \)

so b is the slope at x = 0.