Describing Function Behavior

A graph tells a story, but only if you can put that story into words. Describing function behavior means translating the shape of a curve, or the facts of a real situation, into precise statements about where a function rises, where it falls, where it crosses the axes, and how fast it is changing. This skill matters whenever you read a graph of temperature, distance, profit, or population and need to explain what is actually happening, not just report numbers.

A function is increasing over an interval if its output values get larger as \(x\) gets larger across that interval, and it is decreasing if the output values get smaller as \(x\) increases. A function is constant over an interval if the output does not change at all. When you describe a graph, you always read it from left to right and name the interval of \(x\)-values where each behavior occurs, such as "the function increases for \(x \lt -1\) and decreases for \(-1 \lt x \lt 1\)."

Points where a graph changes from increasing to decreasing, or the reverse, are called turning points. A turning point where the function switches from increasing to decreasing is a relative maximum, and one where it switches from decreasing to increasing is a relative minimum. Naming these turning points is often the clearest way to summarize a graph's overall shape in a sentence.

The y-intercept is the point where a graph crosses the vertical axis, found by evaluating the function at \(x = 0\). The x-intercepts are the points where the graph crosses the horizontal axis, meaning the output equals zero. In a graph with no context, you simply report the intercepts as coordinates. In a graph tied to a real situation, such as height over time, the intercepts carry meaning: the y-intercept might describe a starting height, and an x-intercept might describe the moment the object lands.

Look at the cubic function below. It has a y-intercept at the origin and two other x-intercepts, along with a relative maximum and a relative minimum between them.

Reading this graph from left to right, you can describe it as increasing until the relative maximum, decreasing between the relative maximum and the relative minimum, then increasing again afterward. This kind of sentence, built from the vocabulary of intervals and turning points, is exactly the description this lesson asks you to produce.

Rate of change tells you how quickly a function's output is changing compared to its input. For any interval from \(x = a\) to \(x = b\), the average rate of change is calculated with \( \dfrac{f(b) - f(a)}{b - a} \). A positive result means the function is increasing on average over that interval, a negative result means it is decreasing on average, and a value close to zero means the output barely changes.

Comparing average rate of change across different intervals lets you describe how a function's behavior shifts. For example, a car's distance-time graph might have a rate of change of 60 over one interval and 20 over the next, which you would describe as the car slowing down, even though distance is still increasing in both intervals. This distinction between "increasing" and "increasing quickly versus increasing slowly" is central to a strong verbal description of behavior. For a closer look at pairing this vocabulary with specific graph features, see Interpreting Function Characteristics.

Not every function comes with a picture. A written scenario, such as "a pool is filled at a steady rate for 10 minutes, then drained quickly for 5 minutes," describes function behavior through words alone. You can still identify the same features: an increasing interval, a decreasing interval, an approximate rate of change for each phase, and the point where the behavior switches. Practicing with word scenarios strengthens your ability to move between graphs, tables, and language, since all three are simply different ways of representing the same function.

Increasing interval: A set of \(x\)-values over which the function's output values get larger as \(x\) increases.

Decreasing interval: A set of \(x\)-values over which the function's output values get smaller as \(x\) increases.

Constant interval: A set of \(x\)-values over which the function's output does not change.

Relative maximum: A turning point where the function switches from increasing to decreasing.

Relative minimum: A turning point where the function switches from decreasing to increasing.

x-intercept: A point where the graph crosses the horizontal axis and the output equals zero.

y-intercept: The point where the graph crosses the vertical axis, found at \(x = 0\).

Average rate of change: The ratio \( \dfrac{f(b) - f(a)}{b - a} \), describing how quickly output changes over an interval.

Students describe graphs of everyday situations, such as a hiking trail's elevation or a phone's battery percentage over time, using the vocabulary of increasing and decreasing intervals, intercepts, and rate of change. They also work in reverse, sketching or choosing a graph that matches a written description of behavior.

Related lessons

• Verbal Function Explanation