math 4\nname\ndate\nper\ntest\nfind the amplitude, the period in radians, the phase shift in radians, and…

math 4\nname\ndate\nper\ntest\nfind the amplitude, the period in radians, the phase shift in radians, and the vertical shift.\n1) ( y=\frac{1}{2} cdot sin left(4 \theta-\frac{5 pi}{3}\right)-2 )\n2) ( y=\frac{1}{6} cdot cos left(4 \theta-\frac{7 pi}{4}\right)+2 )\n3) ( y=\frac{1}{9} cdot cos left(5 \theta-\frac{5 pi}{4}\right)+2 )\n4) ( y=6 sin left(3 \theta+\frac{2 pi}{3}\right)+2 )\n5) ( y=-1+\frac{1}{4} cdot cos left(8 \theta-\frac{pi}{3}\right) )\n6) ( y=6 cos left(4 \theta+\frac{7 pi}{4}\right)-3 )\ngraph each function using radians.\n7) ( y=\frac{1}{2} cdot sin \theta )\n8) ( y=2 sin \theta )

math 4\nname\ndate\nper\ntest\nfind the amplitude, the period in radians, the phase shift in radians, and the vertical shift.\n1) ( y=\frac{1}{2} cdot sin left(4 \theta-\frac{5 pi}{3}\right)-2 )\n2) ( y=\frac{1}{6} cdot cos left(4 \theta-\frac{7 pi}{4}\right)+2 )\n3) ( y=\frac{1}{9} cdot cos left(5 \theta-\frac{5 pi}{4}\right)+2 )\n4) ( y=6 sin left(3 \theta+\frac{2 pi}{3}\right)+2 )\n5) ( y=-1+\frac{1}{4} cdot cos left(8 \theta-\frac{pi}{3}\right) )\n6) ( y=6 cos left(4 \theta+\frac{7 pi}{4}\right)-3 )\ngraph each function using radians.\n7) ( y=\frac{1}{2} cdot sin \theta )\n8) ( y=2 sin \theta )

Answer

Explanation:

Step1: Recall the general form of a sinusoidal function

The general form of a sinusoidal function is (y = A\sin(B\theta - C)+D) or (y = A\cos(B\theta - C)+D). The amplitude is (|A|), the period is (T=\frac{2\pi}{|B|}), the phase - shift is (\frac{C}{B}), and the vertical shift is (D).

Step2: Analyze the first function (y=\frac{1}{2}\sin(4\theta-\frac{5\pi}{3}) - 2)

  • Amplitude: (|A|=\left|\frac{1}{2}\right|=\frac{1}{2})
  • (B = 4), so the period (T=\frac{2\pi}{|B|}=\frac{2\pi}{4}=\frac{\pi}{2})
  • (C=\frac{5\pi}{3}), phase - shift (\frac{C}{B}=\frac{\frac{5\pi}{3}}{4}=\frac{5\pi}{12})
  • Vertical shift (D=-2)

Step3: Analyze the second function (y=\frac{1}{6}\cos(4\theta-\frac{7\pi}{4})+2)

  • Amplitude: (|A|=\left|\frac{1}{6}\right|=\frac{1}{6})
  • (B = 4), period (T=\frac{2\pi}{|B|}=\frac{2\pi}{4}=\frac{\pi}{2})
  • (C = \frac{7\pi}{4}), phase - shift (\frac{C}{B}=\frac{\frac{7\pi}{4}}{4}=\frac{7\pi}{16})
  • Vertical shift (D = 2)

Step4: Analyze the third function (y=\frac{1}{9}\cos(5\theta-\frac{5\pi}{4})+2)

  • Amplitude: (|A|=\left|\frac{1}{9}\right|=\frac{1}{9})
  • (B = 5), period (T=\frac{2\pi}{|B|}=\frac{2\pi}{5})
  • (C=\frac{5\pi}{4}), phase - shift (\frac{C}{B}=\frac{\frac{5\pi}{4}}{5}=\frac{\pi}{4})
  • Vertical shift (D = 2)

Step5: Analyze the fourth function (y = 6\sin(3\theta+\frac{2\pi}{3})+2)

Rewrite it as (y=6\sin(3\theta-(-\frac{2\pi}{3}))+2)

  • Amplitude: (|A| = 6)
  • (B = 3), period (T=\frac{2\pi}{|B|}=\frac{2\pi}{3})
  • (C=-\frac{2\pi}{3}), phase - shift (\frac{C}{B}=\frac{-\frac{2\pi}{3}}{3}=-\frac{2\pi}{9})
  • Vertical shift (D = 2)

Step6: Analyze the fifth function (y=-1+\frac{1}{4}\cos(8\theta-\frac{\pi}{3}))

  • Amplitude: (|A|=\left|\frac{1}{4}\right|=\frac{1}{4})
  • (B = 8), period (T=\frac{2\pi}{|B|}=\frac{2\pi}{8}=\frac{\pi}{4})
  • (C=\frac{\pi}{3}), phase - shift (\frac{C}{B}=\frac{\frac{\pi}{3}}{8}=\frac{\pi}{24})
  • Vertical shift (D=-1)

Step7: Analyze the sixth function (y = 6\cos(4\theta+\frac{7\pi}{4})-3)

Rewrite it as (y=6\cos(4\theta-(-\frac{7\pi}{4}))-3)

  • Amplitude: (|A| = 6)
  • (B = 4), period (T=\frac{2\pi}{|B|}=\frac{2\pi}{4}=\frac{\pi}{2})
  • (C=-\frac{7\pi}{4}), phase - shift (\frac{C}{B}=\frac{-\frac{7\pi}{4}}{4}=-\frac{7\pi}{16})
  • Vertical shift (D=-3)

Answer:

  1. Amplitude: (\frac{1}{2}), Period: (\frac{\pi}{2}), Phase - shift: (\frac{5\pi}{12}), Vertical shift: (-2)
  2. Amplitude: (\frac{1}{6}), Period: (\frac{\pi}{2}), Phase - shift: (\frac{7\pi}{16}), Vertical shift: (2)
  3. Amplitude: (\frac{1}{9}), Period: (\frac{2\pi}{5}), Phase - shift: (\frac{\pi}{4}), Vertical shift: (2)
  4. Amplitude: (6), Period: (\frac{2\pi}{3}), Phase - shift: (-\frac{2\pi}{9}), Vertical shift: (2)
  5. Amplitude: (\frac{1}{4}), Period: (\frac{\pi}{4}), Phase - shift: (\frac{\pi}{24}), Vertical shift: (-1)
  6. Amplitude: (6), Period: (\frac{\pi}{2}), Phase - shift: (-\frac{7\pi}{16}), Vertical shift: (-3)