Circular Function Graphs
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Questions and Answers

What is the formula to find the period of the function $y = a \tan b(x - c) + d$?

  • $2b$
  • $\frac{2\pi}{b}$
  • $\frac{\pi}{2b}$ (correct)
  • $\frac{\pi}{b}$
  • In the function $y = 2 \tan 2x$, what values would represent the vertical asymptotes?

  • $x = k\pi, k$ is an odd integer
  • $x = 4 + \frac{k\pi}{2}, k$ is an odd integer
  • $x = \frac{k\pi}{2}, k$ is an odd integer (correct)
  • $x = 2k\pi, k \in \mathbb{Z}$
  • Which of the following is the correct domain for the function $y = 2 \cot \frac{x}{3}$?

  • $x \in \mathbb{R} : x \neq 3k\pi, k \in \mathbb{Z}$ (correct)
  • $x \in \mathbb{R} : x \neq k\frac{\pi}{3}, k \in \mathbb{Z}$
  • $x \in \mathbb{R} : x \neq k\pi, k \in \mathbb{Z}$
  • $x \in \mathbb{R} : x \neq \frac{3k\pi}{2}, k \in \mathbb{Z}$
  • What is the value of $d$ in the function $y = a \tan b(x - c) + d$ given that $a = 1$, $b = 2$, $c = 0$, and $d = 0$?

    <p>0</p> Signup and view all the answers

    For the equation $y = a \cot b(x - c) + d$, which statement about the asymptotes is incorrect?

    <p>The function has no vertical asymptotes if $b$ is negative.</p> Signup and view all the answers

    What is the range of the graph represented by the function $y = -2 ext{sin}(3x + rac{ ext{π}}{3})$?

    <p>(-2, 0)</p> Signup and view all the answers

    What is the amplitude of the function $y = -2 ext{cos}(-2x + rac{ ext{π}}{4})$?

    <p>2</p> Signup and view all the answers

    What is the period of the function $y = 3 ext{sin}(5x - rac{ ext{π}}{2})$?

    <p>$ rac{2 ext{π}}{5}$</p> Signup and view all the answers

    What characteristics define the domain of the function $y = a ext{csc}(b(x - c)) + d$?

    <p>$x ≠ c + rac{k ext{π}}{b}$, where $k$ is an integer</p> Signup and view all the answers

    What is the range of the graph $y = a ext{sec}(b(x - c)) + d$?

    <p>(-∞, d - a) ∪ (d + a, ∞)</p> Signup and view all the answers

    What interval describes the domain of $y = 2 ext{sec}(x)$?

    <p>$x ≠ n rac{ ext{π}}{2}$ for $n$ odd</p> Signup and view all the answers

    What is the amplitude of the function $y = -3 ext{sin}(2x)$?

    <p>3</p> Signup and view all the answers

    Identify the period of the graph $y = - ext{cos}( rac{ ext{π}}{4}(x - 1))$.

    <p>$ rac{8}{ ext{π}}$</p> Signup and view all the answers

    What are the asymptotes of the function $y = - ext{csc}(4x)$?

    <p>$x = rac{k ext{π}}{4}$, where $k$ is any integer</p> Signup and view all the answers

    What is the correct domain for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?

    <p>$x ∈ ℝ: x ≠ \frac{\pi}{8} + k, k ∈ ℤ$</p> Signup and view all the answers

    What is the range for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?

    <p>$(-∞, -3) ∪ (1, ∞)$</p> Signup and view all the answers

    What is the period of the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?

    <p>$\pi$</p> Signup and view all the answers

    Which of the following describes the asymptotes for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?

    <p>$x = \frac{\pi}{8} + k, k ∈ ℤ$</p> Signup and view all the answers

    For the function $y = a cot(b(x - c)) + d$, what represents the domain correctly?

    <p>$x ∈ ℝ: x ≠ c + \frac{k\pi}{b}, k ∈ ℤ$</p> Signup and view all the answers

    Which of the following correctly identifies the amplitude for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?

    <p>2</p> Signup and view all the answers

    What is the correct range for the function $y = a tan(b(x - c)) + d$?

    <p>(-∞, ∞)</p> Signup and view all the answers

    Which of the following statements about the period of $y = a tan(b(x - c)) + d$ is true?

    <p>The period is $\frac{\pi}{|b|}$</p> Signup and view all the answers

    For the function $y = a cot(b(x - c)) + d$, which of these values represents the asymptotes accurately?

    <p>$x = c + \frac{k\pi}{b}, k ∈ ℤ$</p> Signup and view all the answers

    Study Notes

    Circular Function Graphs

    • Graphs of sine, cosine, secant, cosecant, tangent, and cotangent functions are depicted.
    • The domain of sine and cosine functions is all real numbers.
    • The amplitude of y = a sin(b(x - c)) + d and y= a cos(b(x - c)) + d is |a|.
    • The range of y = a sin(b(x - c)) + d and y = a cos(b(x − c)) + d is [d − |a|, d + |a|].
    • The period of y = a sin(b(x - c)) + d and y = a cos(b(x - c)) + d is 2π/|b|.
    • The domain of secant and cosecant functions excludes values where sine and cosine respectively equal zero.
    • The range of secant and cosecant functions is (-∞,-|a|+d] U [|a|+d,∞)
    • The period of secant and cosecant functions is 2π/|b|.
    • The asymptotes of y = a csc(b(x - c)) + d are x = c + (kπ)/|b| where k ∈ Z.
    • The asymptotes of y = a sec(b(x - c)) + d are x = c + (kπ)/(2|b|) where k is an odd integer.
    • The range of tangent and cotangent functions is all real numbers.
    • The period of tangent and cotangent functions is π/|b|.
    • The asymptotes of y =a cot(b(x - c)) + d are x = c + (kπ)/|b| where k ∈ Z.
    • The asymptotes of y = a tan(b(x - c)) + d are x = c + (kπ)/(2|b|) where k is an odd integer.

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    Description

    Explore the properties of various circular functions including sine, cosine, secant, cosecant, tangent, and cotangent. This quiz covers their domains, ranges, periods, and asymptotes. Test your understanding of how these functions behave graphically and mathematically.

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