Increasing and Decreasing Intervals

Pre-Calculus · Axiom Academy

LESSON Increasing and Decreasing Intervals Read where a graph rises and where it falls — and name each stretch with an interval of x-values. 1. Uphill Is Increasing, Downhill Is Decreasing Picture walking along the graph from left to right. Where the ground rises under your feet, the function is increasing : as x grows, y grows too. Where the ground drops away, it is decreasing : as x grows, y shrinks. As x increases, y increases. The graph goes up as you read rightward. As x increases, y decreases. The graph goes down as you read rightward. 2. Turning Points Cut the x-Axis Into Intervals A function only switches between rising and falling at a turning point — the crest of a hill (a local maximum) or the floor of a valley (a local minimum). Drop a line straight down from each turning point to the x -axis, and those x -values slice the axis into the increasing and decreasing intervals. The graph rises until it reaches a local maximum, where it levels off and turns down. The curve is momentarily flat — neither increasing nor decreasing. This x -value is a boundary, not part of either interval. The graph drops until it reaches a local minimum, where it bottoms out and turns back up. The turning points of x^3 - 3x sit at x = -1 and x = 1 — the only places the direction can flip. Walking left to right: the curve rises to the peak at x = -1 , falls down to the valley at x = 1 , then rises again. The turning points at x=-1 and x=1 are exactly where it changes direction.

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