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 $f^{''} (x)$ 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
$f^{''} (x)$	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:

A quintic function with zeros, turning points, and inflection points indicated; below is a breakdown of positive/negative, increasing/decreasing, and concave up/down