Please help! I need to finish this by tonight.
Find the amplitude, period, phase shift, and vertical translation for these two equations:
1. y= -sin(x - pi/4) + 2
2. y= 2cos(2(pi)(x))

Question

whpalmer4 · Answer

Amplitude is half the difference between maximum and minimum values.  For example, the amplitude of $y=\sin x$ is $\frac{1}{2}(1-(-1)) = 1$.  

Period is the smallest interval over which the function repeats.  For $y=\sin x$ the period is $2\pi$.  If you have a function $y = \sin Bx$ the period will be $\frac{2\pi}B$.  

Phase shift is the value added to the argument of the periodic function.  $y=\sin (2x+3\pi)$ has a phase shift of $3\pi$.

Vertical translation is any offset added to the result of the periodic function.  For example, $y=\sin x + 1$ has a vertical translation of 1, and oscillates between 0 and 2 instead of -1 and 1.

whpalmer4 · Answer

Some examples:

First one shows the effects of changing amplitude, phase shift, and translation.  Second one shows changing period, and then a combination of amplitude, phase shift and vertical translation.

anonymous · Answer

Shouldn't the one labeled as 
$$y=2\sin(2x+\frac{ \pi }{ 3 })$$
in sines2.png
be
$$y=2\sin(2x+\frac{ \pi }{ 3 })+1?$$

whpalmer4 · Answer

Nuts, I must have brushed against the delete key after typing it...you're right!

anonymous · Answer

No problem! How did you make those graphs so fast btw?

whpalmer4 · Answer

Mathematica is wonderful, so long as you can figure out how to tell it what you want to do!

anonymous · Answer

ooh, i think i got it.
1. amplitude:1
period:2pi
phase shift: -pi/4
vertical translation: 2

whpalmer4 · Answer

Yes, I believe those are correct.

anonymous · Answer

Alrighty, thanks so much!

whpalmer4 · Answer

How about the second problem?

anonymous · Answer

Oh goodness, how could I forget about that?

anonymous · Answer

Hold on a second.

whpalmer4 · Answer

drum roll please.... :-)

anonymous · Answer

2. amplitude: 2
period: 1

whpalmer4 · Answer

phase shift and vertical translation?

anonymous · Answer

but for phase shift and vertical translation, I'm not so sure.

whpalmer4 · Answer

how about none?  does none strike your fancy?

anonymous · Answer

hmm. none. can you elaborate on that?

whpalmer4 · Answer

There is no phase shift, because you don't have $2\pi x + \phi$ as the argument to $\cos$.  There's no vertical translation, because you don't have anything added on to the result of $2\cos (2\pi x)$ either.

whpalmer4 · Answer

If you had all the options, the formula would look like

$$y = A \sin (Bx + \phi) + C$$ where $A$ controls amplitude, $B$ controls period (or frequency), $\phi$ is the phase shift, and $C$ is vertical translation or offset.

anonymous · Answer

Ahh, that's why I was so utterly stuck in that quagmire. I was overthinking!

anonymous · Answer

Thank you so much!~

whpalmer4 · Answer

It's not quite as helpful as it might be, because it doesn't show you the function you've chosen, but check out http://demonstrations.wolfram.com/SineAndCosineGraphGenerator/ (I think you have to install the Wolfram CDF player, there should be a link on the page) You can play with vertical stretch, phase shift, vertical shift, period with slides and observe the result.

anonymous · Answer

Okie doke! I'll check it out.

whpalmer4 · Answer

Also very interesting if you're curious about possible applications for the magic sin and cos functions appropriately squashed, stretched, shifted, etc. is http://demonstrations.wolfram.com/FourierSeriesOfSimpleFunctions/ Fourier Series allow you to build up just about any waveform by combining the right selection of sines and cosines. This turns out to be an extremely powerful concept in many fields...