Find the period, phase shift, and amplitude of the function. y=3sin(3x-pi/5)+2 Give the exact values, not decimal approximations.

Question

I NEED HELP WITH THIS ASAP

Period:

Phase Shift:

Amplitude:

(These are the things I have to fill out)

Answer 1

Answer 2

For a sine function written in the form y = A sin(B(x − C)) + D, each letter controls exactly one feature of the wave, so the fastest route is to get your function into that shape and then read the answers straight off it.

y = 3 sin(3x − π/5) + 2

The amplitude is the easy one, since it is just the number in front of the sine:

Amplitude = 3

The period comes from the B value. B tells you how much faster the wave cycles than a plain sine does, and a plain sine needs 2π to finish one cycle. With B = 3 you fit three cycles into that same space, so each one takes a third as long:

Period = 2π/3

The phase shift is where almost everyone slips, so it is worth slowing down on. You cannot just read the π/5 that is sitting in there, because the formula wants B factored out of the parentheses first. The inside right now is 3x − π/5, and pulling the 3 out gives:

3x − π/5 = 3(x − π/15)

So your function is really y = 3 sin(3(x − π/15)) + 2, and now the shift is sitting where the formula says to look for it:

Phase Shift = π/15, to the right

If you were about to write π/5, that is exactly the mistake this problem is built to catch. The way I keep it straight is to remember that the horizontal squeeze shrinks the shift along with everything else, so once the graph is compressed by a factor of 3, the shift has to be three times smaller too.

One value you were not asked for but will likely need on the same assignment: the +2 on the end is the vertical shift, so the midline of the wave is y = 2 and the graph runs from y = −1 up to y = 5.

You can check all three answers at once. At x = π/15 the inside of the sine becomes 0, so y = 3 sin(0) + 2 = 2, which puts you exactly on the midline heading upward, which is where a sine wave begins. Go a quarter of a period further, to x = π/15 + π/6, and you should be at the top of the wave, and that does come out to y = 5.

So, our final answers are Period = 2π/3, Phase Shift = π/15 to the right, and Amplitude = 3.