# If we needed to form a trigonometric function to represent the following graph (using sine base), could the phase shift value be simply 1? - Due to the sine base function starting point located at...

If we needed to form a trigonometric function to represent the following graph (using sine base), could the phase shift value be simply 1? - Due to the sine base function starting point located at (0,0), and this graph's start point located at (1,-6.4) (notice how the x-values have gone from 0 to 1); therefore, a phase shift of 1 unit to the right would make sense wouldn't it?

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### 1 Answer

We are given the monthly temperatures and asked to find a trigonometric model (with sine as a base) for the data.

The maximum for the data is about 20.7, the minimum about -6.4 and the period will be 12.

The general form is `y=asin(b(x-h))+k` where a is the amplitude (if a<0 the wave is inverted)((a acts as a vertical stretch/compression)), b affects the period (b acts as a horizontal compression/stretch), h is the horizontal translation (phase shift) and k locates the midline (the vertical shift.)

a: We can find a by `a=("max"-"min")/2` so `a~~13.55`

b: We can find b using `b=(2pi)/p` where p is the period so `b=pi/6`

k: We can find k using `k=("max"+"min")/2` so `k~~7.15`

h: There can be a lot of different numbers that work here. Once you find one, any addition/subtraction of the period will also result in the same model. Adding a half period and changing the sign of a will also work.

The typical sine function begins at the origin. The y value is the value of the midline. So for our model we seek a point in the table (or on the graph) where y=7.15 and the function is increasing. This is a little after April's data point so a good place to start is h=-4. (negative because we are moving the base function to the right. You could also use h=8)

So the phase shift is about 4.

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One model is `y=13.55sin(pi/6x+4)+7.15`

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A graph of the model:

My calculator gives `y=13.55sin(1/2x-2.05)+6.87` as the curve fit. (Note that we took 20.7 for the maximum when it isn't as well as some other short cuts.)