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Graphs of Tangent and Cotangent Functions quiz

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  • What is the identity for the tangent function in terms of sine and cosine?
    The tangent function is defined as tangent(x) = sine(x) / cosine(x).
  • At which x-values do the asymptotes of the tangent graph occur?
    The asymptotes of the tangent graph occur at odd multiples of pi/2, where cosine(x) = 0.
  • What is the period of the basic tangent function?
    The period of the basic tangent function is pi.
  • How do you calculate the period of y = tan(bx)?
    The period is pi divided by b, or period = pi/b.
  • How does the tangent graph behave between its asymptotes?
    The tangent graph increases from bottom left to top right between its asymptotes.
  • Where does the tangent graph cross the x-axis?
    The tangent graph crosses the x-axis at integer multiples of pi.
  • What is the reciprocal identity for cotangent?
    Cotangent is the reciprocal of tangent, so cotangent(x) = 1 / tangent(x).
  • Where are the asymptotes of the cotangent graph located?
    The asymptotes of the cotangent graph are at integer multiples of pi.
  • How does the cotangent graph differ in direction from the tangent graph?
    The cotangent graph descends from top left to bottom right, which is the opposite direction of the tangent graph.
  • What is the period of the cotangent function y = cot(bx)?
    The period is also pi divided by b, or period = pi/b.
  • Where does the cotangent graph cross the x-axis?
    The cotangent graph crosses the x-axis at odd multiples of pi/2.
  • How do the transformation rules for tangent and cotangent compare to those for sine and cosine?
    The transformation rules are the same except the period is pi/b for tangent and cotangent, instead of 2pi/b for sine and cosine.
  • What happens to the tangent and cotangent graphs when b increases in y = tan(bx) or y = cot(bx)?
    As b increases, the period of the graph decreases, causing the curves to repeat more frequently.
  • What is the main difference in the placement of asymptotes between tangent and cotangent graphs?
    Tangent asymptotes are at odd multiples of pi/2, while cotangent asymptotes are at integer multiples of pi.
  • How does the graph of y = tan(pi/2 x) differ from the basic tangent graph?
    Its period is 2, so the curve repeats every 2 units instead of every pi units.