Graph `f(x)=1/3sin(2/3x-pi/4)+4`  Find the amplitude, period, phase shift, and vertical shift of the function.

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Here we have a graph of sin(x) (black) and the given function (blue). Notice how its amplitude is decreased, it is stretched out horizontally, its baseline is elevated, and its intersection with its baseline is pushed forward slightly. Now, we will examine how to determine the actual shifts and stretches.

We will start by getting the function in the canonical form to answer this sort of question, meaning we need to get it to look like the following:

`f(x) = asin(b(x-phi))+c`

We need this form because each parameter in the equation above has to do with each of the things the question asks for. Thankfully, the function is pretty much given in the expected form. We just need to use the distributive property to pull 2/3 out from inside the sine term:

`f(x)=1/3 sin(2/3(x-(3pi)/8)) + 4`

Now, we can solve for each portion of the question.

  • Amplitude: a gives us the amplitude because the sine wave already has a default amplitude of 1. Thus, our amplitude is `1/3`.
  • Period: The default period for a sine wave is `2pi`. The b term is inversely proportional to the period. Therefore, our period is `2pi/(2/3) = 2pi xx 3/2 = 3pi`
  • Phase shift: The horizontal phase shift is given by `phi` in the equation by shifting at what x the sinusoid's input value is equal to zero. Thus, our phase shift is `(3pi)/8`.
  • Vertical shift: Here, c will give us the vertical shift, as it establishes the height above the y-axis is the new "baseline" for the function. Therefore, our vertical shift is 4.

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