1.1 Change in Tandem

Functions describe change in tandem

A function models how two quantities change together: as the input variable changes, the output variable changes in a corresponding way. We describe this using arguments (inputs) and their matching outputs. Analyzing a function means studying how output values respond as input values increase, decrease, or stay constant across an interval.

Increasing, decreasing, and rates of change

A function is increasing on an interval if output values rise as input values increase, and decreasing if outputs fall as inputs increase. The average rate of change over an interval is the change in output divided by the change in input — essentially the slope of the secant line connecting two points on the graph.

Concavity and rate of change behavior

Beyond just increasing/decreasing, we examine whether the rate of change itself is increasing or decreasing. If average rate of change increases across successive intervals, the graph is concave up; if it decreases, the graph is concave down. This helps describe curvature, not just direction.

Key terms

Average rate of change

Change in output divided by change in input over an interval; the slope of the secant line.

Increasing function

Outputs increase as inputs increase over an interval.

Decreasing function

Outputs decrease as inputs increase over an interval.

Concave up

The rate of change is increasing across successive intervals; graph curves upward.

Concave down

The rate of change is decreasing across successive intervals; graph curves downward.

Try an AP-style question

A function is increasing and concave down on an interval. What does that mean?

1. The outputs increase, but at a decreasing rate
2. The outputs decrease, but at an increasing rate
3. The outputs increase at an increasing rate
4. The outputs are positive and getting smaller