I don’t know a truly intuitive way to think about why the axis of symmetry has to be x=-b/2a, but of course once you do know that, you can see how the vertex has to be translated left or right since the vertex has to be on the axis of symmetry.

One thing that can help with understanding the effect of the b-value in the quadratic equation, though, is to think of the quadratic as two combined functions; I’ll call them g(x) and h(x). So if, for example, f(x)=x^2-2x+8, I can also think of f(x) as a combination of g(x)=x^2 + h(x)=-2x+8. In this way, we can see that we’re stacking a parabola g(x)=x^2 on top of a linear function h(x)=-2x+8. If we graph f(x) and h(x), which both have a y-intercept of c, we can see that changing the b-value changes the slope of the line h(x), which is going to cause the parabola to slide left or right. If b is positive, the line has a positive slope (uphill as we read left to right) and the parabola will slide to the downhill to the left. If b is negative, the line has a negative slope (downhill as we read left to right) and the parabola will slide to the downhill to the right. And while I always prefer to think about the a-value making the parabola taller rather than skinnier, it does make the parabola look skinnier and the vertex won’t slide as far to the left or right for a given tilt (slope; b-value). Here’s a desmos page that I threw together to illustrate the ideas; play with the values of a, b, and c, and you’ll see what I mean:
https://www.desmos.com/calc…