Build the Secant Function from Its Graph Quiz

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| Questions: 20 | Updated: Oct 31, 2025
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1) A secant graph has period π, vertical asymptotes at x = −3π/4, −π/4, π/4, 3π/4, … and a minimum value of y = −2 at x = 0.

Explanation

Step 1: Period π ⇒ 2π/|b|=π ⇒ b=2.

Step 2: Minimum y = −2 at x=0 ⇒ negative scale of 2.

So, y = −2 sec(2x).

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About This Quiz
Build The Secant Function From Its Graph Quiz - Quiz

Turn a secant graph into an equation. You’ll read the distance between asymptotes to get the period, use the vertex height to find the vertical scale and shift, and pinpoint the phase shift from the vertex location. Then stitch it all together into a polished model.

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2) The graph of y = sec(x) is shifted right by π/3 and reflected across the x-axis. What is the resulting function?

Explanation

Step 1: Right shift ⇒ (x − π/3).

Step 2: Reflection across x-axis ⇒ negative sign outside.

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3) A secant graph has vertical asymptotes at x = π/6 + kπ/3 and y-intercept (0, 4). What is the function?

Explanation

Step 1: Asymptotes at π/6 + kπ/3 ⇒ 3x = π/2 + kπ ⇒ phase −π/3.

Step 2: Intercept 4 ⇒ scale 4.

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4) Which transformation turns y = sec(x) into y = sec(2x − π)?

Explanation

Step 1: Factor 2 ⇒ horizontal compression by 2.

Step 2: 2x − π = 2(x − π/2) ⇒ shift right π/2.

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5) A graph has vertical asymptotes at x = −π/2, π/2, 3π/2, … and local minima y = 1 at x = 0, 2π, ….

Explanation

Step 1: Asymptotes at ±π/2 match sec(x).

Step 2: Vertex (minimum) 1 at x=0 fits sec(x).

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6) The function y = −3 sec(x − π/4) has which of the following features?

Explanation

Step 1: At x=π/4 the argument is 0 ⇒ sec(0)=1.

Step 2: y=−3 ⇒ on an inverted (maximum) branch.

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7) A secant function has a period of 4π and minimum y = 2 at x = π.

Explanation

Step 1: Period 4π ⇒ b=1/2.

Step 2: Minimum at x=π ⇒ set (x/2 − π/2)=0 ⇒ shift π/2.

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8) A reciprocal trig graph has asymptotes at x = kπ and U-branches centered at x = π/2 + 2πk.

Explanation

Step 1: csc(x) has asymptotes where sin(x)=0 ⇒ x = kπ.

Step 2: Vertices at x = π/2 + 2πk match csc(x).

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9) Which is the correct domain for y = sec(3x − π/2)?

Explanation

Step 1: Denominator zero when cos(3x − π/2)=0 ⇒ 3x − π/2 = π/2 + kπ.

Step 2: Solve ⇒ x = (π/2 + kπ)/3.

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10) The function y = a sec(bx + c) has period π and asymptotes at x = −π/4, π/4.

Explanation

Step 1: Distance between asymptotes is π/2 ⇒ b = 2.

Step 2: Centered at 0 ⇒ c = 0.

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11) A secant curve has asymptotes at x = −π, 0, π, … and a minimum y = −1 at x = −π/2, 3π/2, ….

Explanation

Step 1: Reflecting sec(x) across x-axis gives minima at y = −1.

Step 2: Asymptotes remain at kπ/2 offsets.

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12) Which statement about y = 5 sec(2x) is true?

Explanation

Step 1: Period = 2π/2 = π.

Step 2: Vertical scale 5 ⇒ range ≤ −5 or ≥ 5.

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13) For y = −2 sec(3x + π/3), what is one vertical asymptote?

Explanation

cos(3x + π/3) = 0 ⇒ 3x + π/3 = π/2 ⇒ x = π/18.

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14) A secant function has minimum y = 4 at x = 0 and asymptotes at x = ±π/2.

Explanation

sec(x) has vertex 1 at x=0 ⇒ scaled to 4 gives minimum 4.

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15) Which graph corresponds to y = sec(x) shifted left by π/3 and stretched by 3?

Explanation

Step 1: Left shift by π/3: subtract π/3 from all key x-values.

Step 2: Vertical stretch 3 sets vertex y = ±3.

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16) A graph shows “cup” branches with minima y = 2 at x = π/4 + 2πk, and asymptotes x = −π/4 + πk.

Explanation

Step 1: Minimum at x = π/4 suggests argument zero ⇒ (x − π/4).

Step 2: Scale 2 ⇒ amplitude 2.

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17) For y = a sec(bx), which yields asymptotes every π/6 units?

Explanation

Step 1: Asymptote spacing = π/|b|.

Step 2: π/b = π/6 ⇒ b = 6.

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18) A secant curve has period 2π/5 and smallest positive asymptote at x = π/10.

Explanation

Step 1: Period 2π/b = 2π/5 ⇒ b=5.

Step 2: Asymptotes of sec(5x) at x = π/10 + kπ/5; smallest positive is π/10.

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19) Identify the cosine base of y = sec(2x − π/2).

Explanation

Secant is the reciprocal of cosine ⇒ base cosine is cos(2x − π/2).

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20) A plotted function has asymptotes at x = kπ/2 and a vertex at (0, −1).

Explanation

Standard vertex at (0,1) for sec(x); multiply by −1 ⇒ (0, −1).

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A secant graph has period π, vertical asymptotes at x = −3π/4,...
The graph of y = sec(x) is shifted right by π/3 and reflected across...
A secant graph has vertical asymptotes at x = π/6 + kπ/3 and...
Which transformation turns y = sec(x) into y = sec(2x − π)?
A graph has vertical asymptotes at x = −π/2, π/2, 3π/2, … and...
The function y = −3 sec(x − π/4) has which of the following...
A secant function has a period of 4π and minimum y = 2 at x = π.
A reciprocal trig graph has asymptotes at x = kπ and U-branches...
Which is the correct domain for y = sec(3x − π/2)?
The function y = a sec(bx + c) has period π and asymptotes at x =...
A secant curve has asymptotes at x = −π, 0, π, … and a minimum y...
Which statement about y = 5 sec(2x) is true?
For y = −2 sec(3x + π/3), what is one vertical asymptote?
A secant function has minimum y = 4 at x = 0 and asymptotes at x =...
Which graph corresponds to y = sec(x) shifted left by π/3 and...
A graph shows “cup” branches with minima y = 2 at x = π/4 + 2πk,...
For y = a sec(bx), which yields asymptotes every π/6 units?
A secant curve has period 2π/5 and smallest positive asymptote at x =...
Identify the cosine base of y = sec(2x − π/2).
A plotted function has asymptotes at x = kπ/2 and a vertex at (0,...
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