How can I identify where a function is increasing or decreasing?

Imagine tracing the curve from left to right, as the x-values get larger. If the graph moves upward, its y-values are getting larger, so the function is increasing on that stretch. If the graph moves downward, its y-values are getting smaller, so it is decreasing there. [S1]

For example, a curve that rises as you move from x = 1 to x = 3 is increasing on that interval. The direction is about how the output changes as the input increases—not whether the graph is above or below the x-axis. A curve can be below the axis and still be increasing if it rises from left to right.

Mark the x-values where the graph switches from rising to falling or from falling to rising. Those are the boundaries between stretches to inspect. Then describe the behavior separately on each stretch. A function may decrease, then increase, then decrease again; it does not have to change direction only once. [S1]

Here is a made-up graph, described by its turning points and direction:

x:       -3        -1          2         4
         ●  rises  ●  falls    ●  rises ●
y:      1         4          -2         3

Assume the curve is defined from x = -3 through x = 4 and has no other changes of direction. It increases from x = -3 to x = -1, decreases from x = -1 to x = 2, then increases from x = 2 to x = 4. Using open intervals between the turning points, that is increasing on (-3, -1) and (2, 4), and decreasing on (-1, 2).

The example is illustrative, not a claim about a particular function. If your exercise uses a different domain or specifies a convention for including endpoints, follow that convention.

Scan the entire domain in order, not just the most obvious peak or valley. Each time the direction changes, start a new interval. A small bump can create another interval boundary, so follow the curve carefully and compare its height at nearby x-values.

For a horizontal stretch, the output stays the same over that stretch. It is neither rising nor falling there under the usual classroom description of increasing and decreasing as strict visual directions. Some textbooks use non-strict definitions that allow equality, so check the wording your class uses. OpenStax distinguishes non-strict definitions from strict versions. [S1]

If the graph is too faint or incomplete to tell whether it turns, do not guess. Mark the uncertain x-range and use the equation, a clearer graph, or the problem’s stated assumptions to resolve it.

Yes. Write intervals using x-values, because they identify where along the horizontal axis the behavior occurs. Then attach the behavior word to each interval. For the example above, a clear answer is:

Increasing on (-3, -1) and (2, 4).
Decreasing on (-1, 2).

The y-coordinates of the turning points help you read the graph, but they do not replace the x-intervals in the answer. Before submitting, check that you have covered the full displayed domain, split at every change of direction, and assigned “increasing” or “decreasing” according to the curve’s movement from left to right. The cited definition supports reading the graph this way; it does not determine interval notation conventions for every class or specify how an unseen graph behaves. [S1]