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:

xf(x)110−1−222345678zeroturning pointinflection point+incconcave downupdownupdecincinc+
A quintic function with zeros, turning points, and inflection points indicated; below is a breakdown of positive/negative, increasing/decreasing, and concave up/down