Increasing Function

An increasing function consistently rises as its input increases.

This means that as the x-values grow larger, the y-values of the function also rise.

Visually, the graph of an increasing function increases in y value.

Consider a function that models the altitude of a hot air balloon rising over time. As time progresses, the altitude increases.

Graphically, this function would slope upward, showing a continual gain in height.

The average rate of change over an interval gives a numerical summary of how rapidly the altitude increases.

It is calculated as the change in altitude divided by the change in time over that interval.

Geometrically, the average rate is represented by the slope of the line that connects two points on the graph, known as a secant line.

If the secant line slopes upward, it shows the function gained overall on that interval.

Not all functions increase over their entire domain, but identifying increasing intervals helps analyze growth—like population and carbon emissions.