Curve sketching

When we investigate the behaviour of a function, we try to find out as much as possible about how it behaves. We are interested in things such as monotonicity, that is, whether the function is increasing or decreasing, and on which intervals it is increasing or decreasing. We are also interested in the extrema of the function – its minima and maxima.

Procedure

At the start we are given some function f, typically by a rule such as f(x) = x². Our task is to find out as much as we can about this function. Typically we find:

1. The domain of the function and the range of the function.
2. Whether the function is even or odd.
3. Whether the function is bounded.
4. The points where the graph crosses the x-axis and the y-axis. These are easy to find: we substitute zero for x and work out f(0), which gives the y-intercept. For the x-intercepts we set the whole function equal to zero, f(x) = 0, and find all the roots of the equation.
5. The extrema of the function and the monotonicity of the function.
6. The points of inflection and the intervals of convexity and concavity