Trig Graphs Summary

Pre-Calculus · Axiom Academy

SUMMARY Trig Graphs — Unit Recap Sine, cosine, tangent, and reciprocal graphs — period, amplitude, asymptotes, and the four-parameter transformation that controls them all. and are smooth waves with period , amplitude 1 , and range [-1, 1] about the midline y = 0 . and have period and vertical asymptotes where their denominator is zero. , , are reciprocals — their graphs blow up exactly where the parent function hits zero. The transformation controls amplitude |A| , period , phase shift C , and midline D . Factor B out first so the shift reads as C directly; then plot the 5 key points one quarter-period apart. Starts at the origin, rises to 1 at , returns to 0 at , dips to -1 at , and returns to 0 at . The same wave as sine, shifted left by . Starts at the maximum 1 , then crosses zero at . Core Concept Tangent & Cotangent Vertical asymptotes wherever , at . Each branch climbs from to . Cotangent is the mirror: asymptotes at . Core Concept Reciprocal Graphs Each reciprocal blows up wherever its parent is zero. The range of and is — the U-branches never enter the strip between -1 and 1 . Core Concept The Four-Parameter Transformation One formula encodes every shape-preserving change to a sine or cosine wave. Always factor B out of the argument first — written this way, the horizontal shift reads off directly as C . Mark the start at x = C , then step one quarter-period at a time. Over one cycle the wave traces midline → max → midline → min → midline.

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