Problem

Use the graph of \(y~=~\csc(x)\) on \([0,~2\pi~]\) to state the period, all vertical asymptotes, upward-branch minima, downward-branch maxima, zeros, and range.

Exact csc graph on [0,2π] overlaid with its faint sin source, including every asymptote and branch vertex.

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What this problem is really about

Build the cosecant graph from sine's reciprocal structure. Sine zeros split the graph at vertical asymptotes, while sine's positive and negative unit values become the nearest points of the reciprocal branches. Then use the numerator of the reciprocal and the bound on sine's magnitude to reason about zeros and range, checking every included interval endpoint.

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The period is 2pi; vertical asymptotes are 0, pi, 2pi; upward-branch minima are (pi/2, 1); downward-branch maxima are (3pi/2, -1); there are no zeros; and the range is [-1, 1].

The period is pi; vertical asymptotes are 0, pi, 2pi; upward-branch minima are (pi/2, 1); downward-branch maxima are (3pi/2, -1); there are no zeros; and the range is (-infinity, -1] union [1, infinity).

The period is 2pi; vertical asymptotes are 0, pi, 2pi; upward-branch minima are (pi/2, 1); downward-branch maxima are (3pi/2, -1); there are no zeros; and the range is (-infinity, -1] union [1, infinity).

The period is 2pi; vertical asymptotes are 0, pi, 2pi; downward-branch maxima are (pi/2, 1); upward-branch minima are (3pi/2, -1); there are no zeros; and the range is (-infinity, -1] union [1, infinity).

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Use the graph of \(y~=~\csc(x)\) on \([0,~2\pi~]\) to state the period, all vertical asymptotes, upward-branch minima, downward-branch maxima, zeros, and range.

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