Answer:
We have function,
![y = 3 - 6 \sin {}^{} (2x + \frac{\pi}{2} )](https://tex.z-dn.net/?f=y%20%3D%203%20-%206%20%5Csin%20%7B%7D%5E%7B%7D%20%282x%20%2B%20%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%29%20)
Standard Form of Sinusoid is
![y = - 6 \sin(2x + \frac{\pi}{2} ) + 3](https://tex.z-dn.net/?f=y%20%3D%20%20-%206%20%5Csin%282x%20%2B%20%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%29%20%20%2B%203)
Which corresponds to
![y = a \sin(b(x + c)) + d](https://tex.z-dn.net/?f=y%20%3D%20a%20%5Csin%28b%28x%20%2B%20c%29%29%20%20%2B%20d)
where a is the amplitude
2pi/b is the period
c is phase shift
d is vertical shift or midline.
In the equation equation, we must factor out 2 so we get
![y = - 6(2(x + \frac{\pi}{4} )) + 3](https://tex.z-dn.net/?f=y%20%3D%20%20-%206%282%28x%20%2B%20%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%29%29%20%2B%203)
Also remeber a and b is always positive
So now let answer the questions.
a. The period is
![\frac{2\pi}{ |b| }](https://tex.z-dn.net/?f=%20%5Cfrac%7B2%5Cpi%7D%7B%20%7Cb%7C%20%7D%20)
![\frac{2\pi}{ |2| } = \pi](https://tex.z-dn.net/?f=%20%5Cfrac%7B2%5Cpi%7D%7B%20%7C2%7C%20%7D%20%20%3D%20%5Cpi)
So the period is pi radians.
b. Amplitude is
![| - 6| = 6](https://tex.z-dn.net/?f=%20%7C%20-%206%7C%20%20%3D%206)
Amplitude is 6.
c. Domain of a sinusoid is all reals. Here that stays the same. Range of a sinusoid is [-a+c, a-c]. Put the least number first, and the greatest next.
So using that<em> rule, our range is [6+3, -6+3]= [9,-3] So our range</em> is [-3,9].
D. Plug in 0 for x.
![3 - 6 \sin((2(0) + \frac{\pi}{2} )](https://tex.z-dn.net/?f=3%20-%206%20%5Csin%28%282%280%29%20%2B%20%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%29%20)
![3 - 6 \sin( \frac{\pi}{2} )](https://tex.z-dn.net/?f=3%20-%206%20%5Csin%28%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%29%20)
![3 - 6(1)](https://tex.z-dn.net/?f=3%20-%206%281%29)
![3 - 6](https://tex.z-dn.net/?f=3%20-%206)
![= - 3](https://tex.z-dn.net/?f=%20%3D%20%20-%203)
So the y intercept is (0,-3)
E. To find phase shift, set x-c=0 to solve for phase shift.
![x + \frac{\pi}{4} = 0](https://tex.z-dn.net/?f=x%20%2B%20%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%20%3D%200)
![x = - \frac{\pi}{4}](https://tex.z-dn.net/?f=x%20%3D%20%20-%20%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20)
Negative means to the left, so the phase shift is pi/4 units to the left.
f. Period is PI, so use interval [0,2pi].
Look at the graph above,
1/2 and 1/3 1/4. Is the answer I automatically thought of
The answer is actually 3 and 4.
![\sqrt{7}](https://tex.z-dn.net/?f=%20%5Csqrt%7B7%7D%20)
lies between the two consecutive numbers 3 and 4.
I hope this helped! :))
-3.7 is greater than -3.8, but is it not less than of equal to -3.8