Why does the value of $b$ cause parabola to shift down and left if positive and down and right if negative?

It makes perfect sense why $a$ and $c$ affect the parabola in the ways they do but I don't understand how $b$ has this effect on the graph.

Solution 1:

For some intuition, consider this: for the graph $y=ax^2+bx+c$, the coefficient $c$ denotes the height at which the it intersects the $y$-axis. The coefficient $b$, entirely analogously, denotes the slope at which the graph intersects the $y$-axis.

If $b$ is positive and $a$ is positive, then with the intuition from the above paragraph, it makes complete sense that this makes the parabola move down and to the left, compared to if $b=0$.