Graphing 2D functions

To graph a function of one real number (f(x)), find some points (x, f(x)) on the plane and draw a curve that passes through all of them. When in doubt, find more points to add detail. To make sure you have an accurate curve, try to include these features, if you can find them:

• Where the function is defined
• The limits at ±∞ and on both sides of each discontinuity
• Where the output is positive and where it is negative
• Where the function is increasing, where the function is decreasing, and the outputs of the function where it changes between them (local maximums and minimums)
• Where the derivative of the function is increasing or decreasing, and the outputs of the function where it changes between them:

All diagrams here were made using Geogebra (slightly modified), under CC BY-SA 3.0

If the function has some symmetry (does not change under some transformation), you can graph just part of the function and transform that part to complete the graph. For example, f(−x) = f(x) (reflection over y = 0 has no effect) and −f(−x) = f(x) (τ/2 radian rotation around (0, 0) has no effect) are relatively common.

You can add more detail by finding a simpler approximation for the function around a point (or ±∞). Most of the approximations I’ve seen can be written in this form:

Overall, to graph f(x) = x + 1/(x + 5) + 1/(x − 5), note that:

(I don’t expect you to deal with these calculations, just try to understand the process.)

So f(x) is positive at 0 < x < 4.79, negative at 4.8 < x < 5, and positive at 5 < x.

So f(x) is increasing at 0 < x < 3.99, decreasing at 4 < x < 6, and increasing at 6.01 < x.

So f(x) is concave down at 0 < x < 5, and concave up at 5 < x.