Concavity, Slope, and Graphs

Increasing Function

A function is increasing when its slope is positive, going up as you move right.

Decreasing Function

A function is decreasing when its slope is negative, going down as you move right.

Second Derivative

The second derivative indicates the concavity of a function and helps determine the rate of increase or decrease.

Increase at a Decreasing Rate

The function is going up but doing so more slowly; slope is positive but decreasing, thus the second derivative is negative.

Decrease at an Increasing Rate

The function is going down more steeply; slope is negative but becoming more negative, hence the second derivative is negative.

Increase at an Increasing Rate

The function is going up faster; slope is positive and getting more positive, resulting in a positive second derivative.

Decrease at a Decreasing Rate

The function is going down but slowing its descent; slope is negative but less negative, leading to a positive second derivative.

Concave Up

A graph is concave up if its second derivative is positive, indicating the slopes are increasing.

Concave Down

A graph is concave down if its second derivative is negative, indicating the slopes are decreasing.