Question

Problem A Find the amplitude, the period in radians, the phase shift in radians, and the vertical shift. Then sketch the graph using radians. No computer graphs - sketches need to be on graph paper with properly labeled axes. 1 2) y-2sin 1) y=-2+..cos(30 + 35 4) yaz.cos(30 + )-1 + 1 3) y - 3sin 30+

Answer 1

The general equation of sine or cosine function are

$\\ y= a\sin\left ( bx-c \right )+d\\ \\and \\ \\ y= a\ cos\left ( bx-c \right )+d\\ \\ Where, \\ \\Amplitude=a\\ \\Period= \frac{2\pi}{b}\\ \\Phase \ shift= \frac{c}{b}\\ \\ Vertical\ shift= d\\ \\$

$\\ (1) \\ y= -2+\frac{1}{2}\ cos \left ( 3\theta+\frac{3\pi}{4} \right )\\ \\ Where, \\ \\Amplitude=a=\frac{1}{2}\\ \\Period= \frac{2\pi}{b}=\frac{2\pi}{3}\\ \\Phase \ shift= \frac{c}{b}= -\frac{3\pi}{4}\times\frac{1}{3}= -\frac{\pi}{4} \\ \\ Vertical\ shift= d= -2\\ \\ Graph: \\ \\$

у 4 (1) 3.5 3 2.5 2 1.5 1 0.5 -П -Зry/4 -п/2 -4 0 4 п/2 Зry/4 п -0.5 1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 -5.5 -6 -6.5

(2) y = 2 sin 27 3 - 1 Where, Amplitude = a = 2 Period 27 6 27 1 41 с 27 47 Phase shift = x 2 = 3 Vertical shift=d= -1 Graph:

y 22 (2) 21 20 19 18 17 16 15 14 13 12 11 10 -9 -8 7 6 5 -4 -3 2 1- -40 -8TL/3 -4T1/3 0 4T1/3 8TL/3 4T0 -1 -2 -3 -4 -5 -6 -7 -8- -9 --10- -11- 12 --13 -14 --15 --16. -17 --18 -19 -20 --21- -22 -23| -24

$\\ (3) \\ y=3\sin\left ( 3\theta+\frac{\pi}{2} \right )+1\\ \\ Where, \\ \\Amplitude=a=3\\ \\Period= \frac{2\pi}{b}=\frac{2\pi}{3}\ \ \\Phase \ shift= \frac{c}{b}= -\frac{\pi}{2}\times\frac{1}{3}= -\frac{\pi}{6} \\ \\ Vertical\ shift= d= 1\\ \\ Graph: \\ \\$

у (3) 5 -п -2ry/3 -/3 о 3 23 П -3

$\\ (4) \\ y=\frac{1}{2}\cos\left ( 3\theta+\frac{\pi}{6} \right )-1\\ \\ Where, \\ \\Amplitude=a=\frac{1}{2}\\ \\Period= \frac{2\pi}{b}=\frac{2\pi}{3}\ \ \\Phase \ shift= \frac{c}{b}= -\frac{\pi}{6}\times\frac{1}{3}= -\frac{\pi}{18} \\ \\ Vertical\ shift= d=- 1\\ \\ Graph: \\ \\$

y (4) 2 -1 -11/2 -TI/3 -TI/6 0 T/6 TI/3 TI/2 -2