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: