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Before, whenever I did direction fields I used equations with x and y, and graphed how it looked all over.

Now I'm supposed to work with direction fields (in Differential Equations) that have the variables $t$ and $y$. Should I start graphing only after $t=0?$ Conceptually, negative times don't really make sense, but the examples we're using don't seem to have any bounds. It seems pretty arbitrary to use t instead of $x$ right now since we're not relating it to real life, and also silly to start at $t=0$.

Is it just convention to start at $t=0$ whenever you're using that notation, even if its not related to real life and isn't a real bound?

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    If your $t=0$ represents the time when the Big Bang occurred, then indeed it makes little sense to talk of negative $t.$ Otherwise, plenty of things happened before $t=0$ and you can use negative $t$ to describe them. I don't see a conceptual problem at all. -- And even the Big Bang example assumes that $t$ represents time, which is not always true in differential equations. – 2017-02-10
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    An example where $t<0$ would not make sense is tossing a ball into the air: the acceleration is $g=9.8\text{ m/s}^2$ downwards, but only after you throw it! (For that matter, that acceleration will cease to be valid once the ball hits the floor). On the other hand, you could just as readily redefine $t=0$ to be when the ball is at its maximum height, and in that case negative times are as valid as positive ones. – 2017-02-10

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