Secant Branch Behavior Quiz: Secant Branch Behavior Between Asymptotes

secθ > 0 exactly when cosθ > 0, which happens in Quadrants I and IV. In [0,2π): (0,π/2) and (3π/2,2π). The combined description in A is equivalent.

Extrema occur at θ = kπ; asymptotes occur at θ = π/2 + kπ. Each asymptote sits exactly midway between two neighboring extrema.

sec(θ+2π)=1/cos(θ+2π)=1/cosθ=secθ, and no smaller positive shift repeats the whole graph everywhere. Hence the period is 2π.

Upward-opening branches occur where cosθ>0: around 0 and around 2π. Those are (−π/2, π/2) and (3π/2, 5π/2). Intervals spanning mixed quadrants are not single branches.

secθ has local minima where cosθ = 1, i.e., θ = 2kπ. Within [−2π,2π], that gives θ = −2π, 0, 2π.

Because |cosθ| ≤ 1, its reciprocal satisfies |secθ| ≥ 1. Thus the range is all real numbers with magnitude at least 1.

Within an interval between consecutive asymptotes, secθ is smooth and attains its unique extremum at the midpoint (θ = kπ), with value ±1 depending on the sign of cos.

Because |cosθ| ≤ 1, its reciprocal satisfies |secθ| ≥ 1. Each branch covers either y ≥ 1 (upward) or y ≤ −1 (downward).

Vertices occur at θ=kπ with y=±1. Each branch lies between consecutive asymptotes. sec never crosses y=0. Center at 2kπ gives cos=1 so it opens up; center at (2k+1)π gives cos=−1 so it opens down.

Near π/2, cosθ→0 with sign change: from the left cosθ→0^+ so secθ=1/cosθ→+∞; from the right cosθ→0^− so secθ→−∞.

At θ = kπ, cosθ = ±1 so secθ = ±1. These occur midway between asymptotes with derivative zero, giving local minima at even k (value 1) and local maxima at odd k (value −1).

secθ < 0 when cosθ < 0, which is Quadrant II and III together: (π/2, 3π/2). On this interval the branch opens downward with a maximum at θ = π.

On each open interval (π/2 + kπ, π/2 + (k+1)π), secθ is continuous and takes either all values ≥ 1 or ≤ −1. The branch is the reciprocal of a cosine half-wave, forming one smooth arc.

A and C are definitional. B follows since |cosθ| ≤ 1 ⇒ |1/cosθ| ≥ 1. The branch opens upward when secθ>0, i.e., cosθ>0; when cosθ<0 it opens downward.

Since secθ = 1/cosθ, it is undefined exactly at the zeros of cosine, θ = π/2 + kπ.

Upward-opening branches occur where cos>0, between asymptotes at −π/2 and π/2, centered at θ=0. Between π/2 and 3π/2 the branch opens downward.

On (−π/2, π/2), cosθ > 0 so secθ > 0, giving a U-shaped arc with its minimum at θ=0 where sec0=1. No asymptote at 0 and sec never crosses 0.

In (π/2,3π/2), cosθ < 0 so secθ < 0. The reciprocal of the negative cosine half-wave yields an inverted U with a peak at θ = π where secπ = −1.

cos is even, so secθ = 1/cosθ is also even: sec(−θ)=1/cos(−θ)=1/cosθ=secθ. This symmetry mirrors branches across the y-axis.

secθ = 1/cosθ is undefined when cosθ = 0, which occurs at θ = π/2 + kπ for any integer k. These are the graph’s vertical asymptotes.