# Graphing

If you want to have a complete graph of y = f(x), then these are all of the things that you should make sure show up:
• x = 0, if f(0) exists;
• x → ∞, if f(x) exists in that direction;
• x → −∞, if f(x) exists in that direction;
• x → c, if f(x) exists in that direction, whenever f is undefined or discontinuous at c;
• x → c+, if f(x) exists in that direction, whenever f is undefined or discontinuous at c;
• x = c, if f(c) exists, whenever f is undefined approaching c from either direction (or both);
• x = c, whenever f(c) = 0;
• x = c, if f(c) exists, whenever f′ is undefined or discontinuous at c;
• x = c, whenever f′(c) = 0;
• x = c, if f(c) exists, whenever f′′ is undefined or discontinuous at c;
• x = c, whenever f′′(c) = 0.
This should be sufficient whenever f is a twice-differentiable function whose domain is an interval, or more generally whenever f is piecewise twice-differentiable: a piecewise-defined function in which the domain of each piece is an interval and in which each piece is twice-differentiable except possibly at its endpoints. (There are weirder functions that can't be put in this form, but you shouldn't have to deal with them in this class.)

If you have a graphing calculator, then you may use it, but you still need to ensure that all of the features listed above appear. At the very least, this may require you to adjust the calculator's graphing window. If you're graphing by hand, then you'll get the best results if you know the values or limits of f, f′, and f′′ for all of these places or limits, but you should at least get f for all of them and f′ whenever you looked there because of something involving f′ or f′′. You can also look at places in between these (if f is defined there) for an even more precise graph.

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