Problem

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

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

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What this problem is really about

View secant as the reciprocal of cosine and let the cosine graph organize every feature. Its zeros create breaks and vertical asymptotes, while its positive and negative unit extrema remain fixed as reciprocal-branch vertices. The inherited repeat and symmetry handle period and parity, and the closed viewing interval means both endpoints must be checked.

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

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

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

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

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