2.2.13 Graphs of Sines/Cosines

Popular questions for the last column involve determining amplitudes, periods, phase shifts, and vertical shifts for plots of sines/cosines.

The general equation for any sine/cosine plot is: y=Asin⁡[B(x−C)]+Dy = A \sin[B(x - C)] + D

| Property | Formula | | ------------------ | ---------------------------- | --- | --- | | Amplitude | ∣A∣ | A | | | Period | 2πB\frac{2\pi}{B} | | Phase Shift | CC | | Vertical Shift | DD (Up if > 0, Down if < 0) |

Example: Find the period of y=3sin⁡(πx−2)+8y = 3 \sin(\pi x - 2) + 8. We need the coefficient in front of xx to be 1, so we need to factor out π\pi: y=3sin⁡[π(x−2π)]+8y = 3 \sin[\pi(x - \frac{2}{\pi})] + 8

  • Period = 2π/π=22\pi / \pi = 2
  • Amplitude = 3
  • Phase Shift = 2/π2/\pi
  • Vertical Shift = 8

Problem Set 2.2.13

What is the amplitude of y = 4 cos(2x) + 1
The graph of y = 2 - 3 cos[2(x - 5)] has a horizontal displacement of
The graph of y = 2 - 2 cos[3(x - 5)] has a vertical shift of
What is the amplitude of y = 2 - 3 cos[4(x + 5)]
The period of y = 5 cos(1/4(x + 3π)) + 2 is kπ, k =
The phase shift of y = 5 cos[4(x + 3)] - 2 is
The amplitude of y = 2 - 5 cos[4(x - 3)] is
The vertical displacement of y = 5 cos[4(x + 3)] - 2 is
The phase shift of f(x) = 2 sin(3x - π/2) is kπ, k =
The period of y = 2 - 3 cos(4πx + 2π) is
The period of y = 2 + 3 sin(x/5) is
The graph of y = 1 - 2 cos(3x + 4) has an amplitude of