## Value, slope, & concavity

When sketching the graph of a function, the obvious first step is to plot a few points. Specifically, the zeros, the turning points, and the inflection points (and, for good measure, the y-intercept). But how does one connect the dots? The answer lies within three quantities: function value, slope, and concavity. In fact, it’s just their signs that matter.

The values of f(x), f'(x), and fʺ(x) represent function value, slope, and concavity, respectively. This table summarizes the meaning of their signs:

Value = 0 > 0 < 0
f(x) zero positive negative
f'(x) turning point increasing decreasing
fʺ(x) inflection point concave up concave down

Here is an example of the information that these three quantities, given by the function and its first two derivatives, provides you with: