Transforming the Tangent Function

Tangent's graph is built from asymptotes, not from a highest or lowest point, and that's the key to understanding its transformations. Asymptotes occur where cosine equals zero, since tan(x) = sin(x)/cos(x). A vertical stretch, the a in y = a·tan(x), multiplies every output value, making branches rise and fall more steeply. But multiplying outputs by a nonzero number never turns a zero denominator into something else, so the asymptotes stay exactly where they were. If a is negative, each branch flips across the x-axis: what used to climb from bottom-left to top-right now falls from top-left to bottom-right, but the vertical lines marking the asymptotes don't budge.

The b inside y = tan(bx) works differently, because it rescales the input before cosine ever sees it. Ordinary tangent has period π, with asymptotes at x = π/2 + k·π. Replace x with bx, and the whole pattern compresses or stretches horizontally by a factor of 1/b, giving a new period of π/b and squeezing or spreading the asymptotes to match. Larger b packs branches closer together; smaller b spreads them out.

Shifts work just like they do for any function: c inside the parentheses slides the graph left or right, and d outside slides it up or down. These four moves combine the same way they do for sine, cosine, and every other function family you've stretched and shifted this unit, which means tangent's transformations aren't a new idea, just a familiar one applied to a trickier-looking parent graph.