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Graphs and Variations of Trigonometric Functions

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Termini in questo insieme (20)

  • Domain of sine and cosine functions
    The domain is (-∞, ∞), all real numbers. The graph extends indefinitely left and right with no gaps or holes.
  • Range of sine and cosine functions
    The range is [-1, 1], the set of all real numbers between -1 and 1 inclusive.
  • Period of sine and cosine functions
    The period is \(2\pi\). The graph's pattern repeats every interval of \(2\pi\).
  • Odd function property of sine
    Sine is an odd function: \(\sin(-x) = -\sin x\). Its graph is symmetric with respect to the origin.
  • Key points for graphing y = sin x
    At \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\), y-values are 0, 1, 0, -1, 0 respectively.
  • Graph behavior of y = sin x between key points
    From 0 to \(\frac{\pi}{2}\), y increases from 0 to 1; from \(\frac{\pi}{2}\) to \(\pi\), y decreases from 1 to 0.
  • Amplitude of y = A sin(Bx - C)
    Amplitude is the absolute value of A: \(|A|\).
  • Period of y = A sin(Bx - C)
    Period is \(\frac{2\pi}{B}\).
  • Phase shift of y = A sin(Bx - C)
    Phase shift is \(\frac{C}{B}\). If positive, shift right; if negative, shift left.
  • Graphing variations of y = sin x
    Identify amplitude and period, find key points, plot values, connect points with smooth curve, and extend graph as needed.
  • Key points for graphing y = cos x
    At \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\), y-values are 1, 0, -1, 0, 1 respectively.
  • Graph behavior of y = cos x between key points
    From 0 to \(\frac{\pi}{2}\), y decreases from 1 to 0; from \(\frac{\pi}{2}\) to \(\pi\), y decreases from 0 to -1.
  • Graph of y = A cos(Bx - C)
    Amplitude is \(|A|\), period is \(\frac{2\pi}{B}\), and phase shift is \(\frac{C}{B}\).
  • Domain of tangent and cotangent functions
    Domain is all real numbers except odd multiples of \(\frac{\pi}{2}\) for tangent, and multiples of \(\pi\) for cotangent.
  • Range of tangent and cotangent functions
    Range is all real numbers (-∞, ∞).
  • Period of tangent and cotangent functions
    Period is \(\pi\).
  • Graphing y = tan(x + π/4)
    Shift the basic tangent graph left by \(\frac{\pi}{4}\) and graph two full periods.
  • Domain of secant and cosecant functions
    Domain is all real numbers except where cosine or sine equals zero, causing vertical asymptotes.
  • Range of secant and cosecant functions
    Range is (-∞, -1] ∪ [1, ∞).
  • Period of secant and cosecant functions
    Period is the same as their base functions: \(2\pi\).