As far as when to use technology for graphs and when to do it by hand, I think in a sense you answered it. If you’re just using integer values, like with most lines, or factored form, you should do it by hand. You get to see the relationships and connections “first hand”. If your roots or axis of symmetry is going to be irrational or decimalized, head to the calculator or equivalent.
This might even be a good question to toss out at student. In fact, why is x^2 + 3 so much easier to graph than x^2 + x + 3? At what point do you find graphing challenging? Hm, maybe that’s a better answer – go to the technology when you start to think a graph is more tedious than challenging. Meaning it may be different for everyone. But then that’s just my thoughts.

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