# 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 $f(x)\text{,}$ ${f}^{\prime}(x)\text{,}$ and ${f}^{\prime \prime}(x)$ represent function value, slope, and concavity, respectively. This table summarizes the meaning of their signs:

Value | $=0$ | $>0$ | $<0$ |
---|---|---|---|

$f(x)$ | zero | positive | negative |

${f}^{\prime}(x)$ | turning point | increasing | decreasing |

${f}^{\prime \prime}(x)$ | 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: