Trig Function Graph

.I literally just watched a video on how to do this, this is a learning process for me too 

I am assuming that you learned the trig vocabulary for this as you asking these problems

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Lets first do a cosine graph cuz its easier

This is the base equation for a cosine graph:

\(y=A\cos{b}(x-h)+c\)

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So first find the amplitude, which is the height of the waves of the graph. (using y-values)

\(\text{Amplitude}=|\frac{\text{Max}-\text{Min}}{2}|\)

\(|\frac{-5-(-4)}{2}|\)

\(|\frac{-1}{2}|\)

\(\text{Amplitude}=\frac{1}{2}\)

Now we have the "A" value

\(y=\frac{1}{2}\cos{b}(x-h)+c\)

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Ok now we have to find the period

We first find the positive difference of the x values to find the horizontal distance.

\(|-1-3.5|=4.5\)

Then we double what we got

\(9\)

Then we solve for the b-value

\(9=\frac{2\pi}{b}\rightarrow9b=2pi\rightarrow{b}=\frac{2pi}{9}\)

Now we have

\(y=\frac{1}{2}\cos{\frac{2pi}{9}}(x-h)+c\)

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Now we have to find the phase shift (h-value)

Since it has shifted -1,                          ( coordinate (-1, -5) tells us that. )

We now have:

\(y=\frac{1}{2}\cos{\frac{2pi}{9}}(x+1)+c\)

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Now we have to find the vertical shift (c-value)

Formula for that is 

\(c=\frac{\text{Maximum}}{2}\)

\(c=\frac{-5+(-4)}{2}\)

\(c=-4.5\)

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So the equation of the sinusoidal graph is

\(y=\frac{1}{2}\cos{\frac{2pi}{9}}(x+1)-4.5\)

This is the cosine graph.

This is the video I learned from, if you want to find the sine graph, follow the steps in the video.