Help With Sinusoidal Equation

Question

How do I find the equtation for the Sinusoidal Equation in BLUE?

Answer 1

You have a choice of using either sine or cosine, both of which are sinusoidal graphs.

The basic equation of an untranslated sine graph in terms of its amplitude $A$ and period $p$ is

$$ y=A\sin\left(\frac{2\pi}{p}x\right)$$

But if it is translated horizontally by $h$ units and vertically by $k$ units the translated equation becomes

$$ y-k=A\sin\left(\frac{2\pi}{p}(x-h)\right)$$

which us usually written in the form

$$ y=A\sin\left(\frac{2\pi}{p}(x-h)\right)+k\tag{1}$$

To obtain the equation by visual inspection of a sinusoidal graph one must estimate the values of the amplitude and period and the amount of vertical or horizontal translation.

The amplitude is half the vertical distance between the peaks and troughs so we can conclude that $A=1$. There apears to be no vertical shift so we can conclude that $k=0$. Close inspections reveals a small horizontal shift of an estimated amount of $h=\frac{\pi}{24}$. The period is the horizontal distance from peak to peak which appears to be $p=\frac{3\pi}{2}$. Substituting these values into equation $(1)$ obtains

$$ y=\sin\left(\frac{4}{3}\left(x-\frac{\pi}{24}\right)\right)$$

which may be simplified to the form

$$ y=\sin\left(\frac{4}{3}x-\frac{\pi}{18}\right)$$