the amplitude is (max -min)/2 = 13-1/2=6
is b=2pi/period , the period seems to be about 6, so b=pi/3
c = horizontal shift(I doesn't seem to be a horizontal shift). How do I calculate it?
d= (max+min)/2 = 13 + 1/2=14/2=7
Dear terminator,
According to the details you have found,
$\displaystyle y=6\sin\{\frac{\pi}{3}\left(x+c\right)\}+7$
When $\displaystyle x=0\Rightarrow y=7$
$\displaystyle 7=6\sin\{\frac{\pi}{3}\left(c\right)\}+7\Rightarro w \dfrac{\pi c}{3}=n\pi~where~n\in Z$
$\displaystyle c=3n~where~n\in Z$
Hope this will help you.
It's also true that $\displaystyle \displaystyle y'>0$ at $\displaystyle \displaystyle x=0$
so, $\displaystyle \displaystyle {{\pi c}\over{3}}\approx 2\pi n$.
Thus: $\displaystyle \displaystyle c\approx 6 n$.
Since the simplest case of $\displaystyle \displaystyle c\approx 6 n$ is when $\displaystyle \displaystyle n=0$, I would choose $\displaystyle \displaystyle c\approx 0\ .$
Also, it looks like the period is closer to 6.2 than it is to 6, (BTW: That's very close to $\displaystyle \displaystyle 2\pi\ .$) so I would say that $\displaystyle \displaystyle b\approx {{\pi}\over{3.1}}\ ,$ maybe even $\displaystyle \displaystyle b\approx 1\ .$