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it is infinite and is a parabola

Its first derivative is non-zero

* * * * *

That is not a proper answer. All that it means is that the relationship between the two variables in question (x and y) is some nonlinear continuous function.

  1. It need not be infinite - a graph can be a circle: that is curved but is certainly not infinite. It could also be an ellipse, a hyperbola, a polynomial of any degree other than 1, an exponential or logarithmic curve, a trigonometric curve, or any combination of the above - as well as other curves.
  2. The point about the first derivative is true, but that also applies to any straight line that is not horizontal.

Actually the first derivative can be zero at specific point(s), but this is specifically true whereas a non-zero value is generally true.

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Q: What does it mean when the line is curved on a graph?
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