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Questions and Answers
What is the formula to find the period of the function $y = a \tan b(x - c) + d$?
What is the formula to find the period of the function $y = a \tan b(x - c) + d$?
In the function $y = 2 \tan 2x$, what values would represent the vertical asymptotes?
In the function $y = 2 \tan 2x$, what values would represent the vertical asymptotes?
Which of the following is the correct domain for the function $y = 2 \cot \frac{x}{3}$?
Which of the following is the correct domain for the function $y = 2 \cot \frac{x}{3}$?
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 $d = 0$?
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 $d = 0$?
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For the equation $y = a \cot b(x - c) + d$, which statement about the asymptotes is incorrect?
For the equation $y = a \cot b(x - c) + d$, which statement about the asymptotes is incorrect?
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What is the range of the graph represented by the function $y = -2 ext{sin}(3x + rac{ ext{π}}{3})$?
What is the range of the graph represented by the function $y = -2 ext{sin}(3x + rac{ ext{π}}{3})$?
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What is the amplitude of the function $y = -2 ext{cos}(-2x + rac{ ext{π}}{4})$?
What is the amplitude of the function $y = -2 ext{cos}(-2x + rac{ ext{π}}{4})$?
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What is the period of the function $y = 3 ext{sin}(5x - rac{ ext{π}}{2})$?
What is the period of the function $y = 3 ext{sin}(5x - rac{ ext{π}}{2})$?
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What characteristics define the domain of the function $y = a ext{csc}(b(x - c)) + d$?
What characteristics define the domain of the function $y = a ext{csc}(b(x - c)) + d$?
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What is the range of the graph $y = a ext{sec}(b(x - c)) + d$?
What is the range of the graph $y = a ext{sec}(b(x - c)) + d$?
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What interval describes the domain of $y = 2 ext{sec}(x)$?
What interval describes the domain of $y = 2 ext{sec}(x)$?
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What is the amplitude of the function $y = -3 ext{sin}(2x)$?
What is the amplitude of the function $y = -3 ext{sin}(2x)$?
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Identify the period of the graph $y = - ext{cos}(rac{ ext{π}}{4}(x - 1))$.
Identify the period of the graph $y = - ext{cos}(rac{ ext{π}}{4}(x - 1))$.
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What are the asymptotes of the function $y = - ext{csc}(4x)$?
What are the asymptotes of the function $y = - ext{csc}(4x)$?
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What is the correct domain for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?
What is the correct domain for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?
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What is the range for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?
What is the range for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?
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What is the period of the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?
What is the period of the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?
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Which of the following describes the asymptotes for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?
Which of the following describes the asymptotes for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?
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For the function $y = a cot(b(x - c)) + d$, what represents the domain correctly?
For the function $y = a cot(b(x - c)) + d$, what represents the domain correctly?
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Which of the following correctly identifies the amplitude for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?
Which of the following correctly identifies the amplitude for the function $y = -2 csc(-2x - \frac{\pi}{4}) - 1$?
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What is the correct range for the function $y = a tan(b(x - c)) + d$?
What is the correct range for the function $y = a tan(b(x - c)) + d$?
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Which of the following statements about the period of $y = a tan(b(x - c)) + d$ is true?
Which of the following statements about the period of $y = a tan(b(x - c)) + d$ is true?
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For the function $y = a cot(b(x - c)) + d$, which of these values represents the asymptotes accurately?
For the function $y = a cot(b(x - c)) + d$, which of these values represents the asymptotes accurately?
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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.