Secant Branch Behavior Quiz: Secant Branch Behavior Between Asymptotes

1) The points θ = kπ are local extrema of y = secθ (where defined).

True

False

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).

Explanation

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).

2) Where are the vertical asymptotes of y = secθ?

θ = π/2 + kπ

θ = kπ

θ = 2π + kπ

No vertical asymptotes

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.

Explanation

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.

3) The graph of y = secθ is even: sec(−θ) = secθ.

True

False

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.

Explanation

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.

4) Which description matches the branch on (π/2, 3π/2)?

Opens downward with maximum y = −1 at θ = π

Opens upward with minimum y = 1 at θ = π

Opens upward and crosses y = 0

Opens downward and crosses y = 0

In (π/2,3π/2), cosθ

Explanation

In (π/2,3π/2), cosθ

5) Select all true statements about the branch on (−π/2, π/2).

Secθ > 0

Branch opens upward

Minimum occurs at θ = 0 with value 1

There is a vertical asymptote at θ = 0

The branch crosses y = 0

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.

Explanation

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.

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6) Which pair correctly lists consecutive asymptotes enclosing an upward-opening branch?

θ = −π/2 and θ = π/2

θ = 0 and θ = π

θ = π/2 and θ = 3π/2 (upward)

θ = π and θ = 2π

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.

Explanation

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.

7) Write the domain of y = secθ in words.

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

Explanation

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

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8) Select all correct relationships between sec and cos that help decide branch opening.

Secθ and cosθ have the same sign

|secθ| ≥ 1 whenever defined

Secθ = 1/cosθ

If cosθ < 0 then the sec-branch opens upward

If cosθ > 0 then the sec-branch opens upward

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

Explanation

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

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9) Between any two consecutive asymptotes, a branch of y = secθ is a single U-shaped or inverted U-shaped arc.

True

False

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.

Explanation

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.

10) On which interval does a branch of y = secθ open downward (secθ < 0)?

(π/2, 3π/2)

(0, π/2)

(3π/2, 2π)

(−π/2, π/2)

secθ

Explanation

secθ

11) Select all intervals on [0, 2π) where a secθ branch opens upward (secθ > 0).

(−π/2, π/2) shifted into [0,2π) → (0, π/2)∪(3π/2, 2π)

(π/2, 3π/2)

(0, π/2)

(π, 3π/2)

(3π/2, 2π)

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.

Explanation

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.

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12) Complete: As θ → (π/2)^−, secθ → ____ and as θ → (π/2)^+, secθ → ____.

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

Explanation

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

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13) Select all accurate sketches/descriptions for a branch of y = secθ.

Has a vertex at (kπ, ±1)

Is confined between two vertical asymptotes at θ = π/2 + kπ and θ = π/2 + (k+1)π

Crosses the x-axis once per branch

Opens upward when centered at θ = 2kπ

Opens downward when centered at θ = (2k+1)π

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.

Explanation

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.

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14) Give the y-values that secθ can take on any branch.

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

Explanation

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

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15) Each branch of y = secθ has exactly one extremum: a minimum if the branch opens up, or a maximum if it opens down.

True

False

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.

Explanation

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.

16) Complete: The range of y = secθ is ____.

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

Explanation

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

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17) At which θ does the branch minimum y = 1 occur in [−2π, 2π]?

θ = 2kπ for k ∈ {−1, 0, 1}

θ = (2k+1)π for k ∈ {−1, 0, 1}

θ = π/2 + kπ for k ∈ {−2, −1, 0, 1, 2}

None of these

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

Explanation

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

18) Select all intervals that together show two consecutive upward-opening branches of y = secθ.

(−π/2, π/2)

(π/2, 3π/2)

(3π/2, 5π/2)

(π, 2π)

(0, π)

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.

Explanation

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.

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19) State the fundamental period of y = secθ.

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.

Explanation

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.

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20) Vertical asymptotes of y = secθ occur halfway between consecutive extrema.

True

False

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

Explanation

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