In a quadratic function:
coefficient $a$ controls the speed of increase/decrease from the vertex.
coefficient $b$ controls the downward slope as the function crosses the y-axis.
I don't really get what the coefficient $b$ is doing. Is it just controlling the steepness of the slope after it crosses the y-axis? Isn't that what $a$ is doing by controlling the speed of increase/decrease (i.e., controlling the steepness/slope of the graph and how "closed" it is)?
Why should the slope change as it crosses the y-axis?
I guess I'm not really seeing the difference between $a$ and $b$.
If possible, in addition to an explanation, could you provide a picture of a parabola (quadratic function) and point to what part of the graph is being determined by $a$ and $b$?