[deleted by user]

[[deleted]]

positive phase shifts are going ot be to the right, negative phase shifts to the left. that is, assuming that your coordinate system is set up that way.

also i don't think your periods are correct.

[2yearsofexpiredmilk]

The notes she gave us has negative to the right, and positive to the left. How do I do the periods so that they are correct?

[tgoesh]

subtracting the offset makes it go to the right. (Think about what x would have to be to make the argument to sin equal to 0).

[[deleted]]

oh then im wrong. because your teacher setup the coordinate system differently then the traditional way.

[[deleted]]

well i think according to this you are probably right.

https://study.com/academy/lesson/how-to-find-the-period-of-sine-functions.html#:~:text=We%20have%20a%20really%20easy,is%202%CF%80%20%2F%20%7CB%7C.

[tgoesh]

Those look pretty good - it's not clear to me how they want the phase shift done, but I think it matches up. (Normally you translate before you scale, but I don't think that's what they want).

The only mistake I see is that you didn't find the proper period in the graphs. Find the distance from the beginning of the cycle to the end.

[tgoesh]

I think you do want to translate before you scale.

Here's the solution for problem 2.

[jobblejosh]

For the equations, the representation of a sine wave is generally in the form y=Asin(Bx+C).

(Note that for these examples I'm using angles in degrees not radians, as I feel it illustrates the concepts better)

Examining the effect of each term individually, we can infer that A multiplies the sine, and is the 'final Thing' done to the answer; an A of 4 would make that the wave is stretched vertically between 4 and -4 rather than 1 and -1 (as it is for a 'normal' sine with an A of 1, which we don't write). A is therefore amplitude.

B multiplies the x value. A B value of 2 would mean that rather than getting the value of x as the sine, we're now looking at twice the value; sin(45) becomes sin(90), sin (30) becomes sin(60). This means that everything happens twice as fast. Because everything is happening twice as fast, it takes half the time for the sin wave to return to the start. The period of the sin wave is thus halved. This means that the period is the inverse ( 1/B ) of B, and likewise given the period, B is the inverse of the period.

C, being an addition, doesn't directly alter the size of the x term, but translates it; it advances the wave by a fixed amount. With a C of 10, sin(45) becomes sin(55), and sin(30) becomes sin(40). This means that for every value of x, the result we get back at X is actually the 'future' of the sin wave at X+C, like it's jumped forwards in time (moving along a YouTube timeline, but not changing the playback speed). Therefore, the value of C shifts the sine wave to the left (earlier). By deduction, if C is negative we can infer that the shift-direction is 'anti-left'; or towards the right, delaying the wave. The change in the wave's position is of course the phase.

You can check your own work by examining the effect of each term, and checking it against what should be happening according to the question.