Problem: May 7th is a full moon (meaning we see all of the moon). A full moon happens every 29.5 days (I know, it’s kind of weird that it’s a half day, just go with it); halfway in between is a new moon (we see none of the moon). Write an equation (and graph it) to model the percentage of the moon that we will see on any given day starting with today as day zero. What percent of the moon will we see today? FYI Today is April 21st.
What I've tried: I'm not sure wether it's a sine or cosine graph, but I think it would be the former because with a cosine graph, the halfway point is at -1, which doesn't make sense because I need to find the percentage. So I think I got a somewhat basic equation:
$$y=\sin \left(\frac{2\pi }{29.5}x\right)$$
But now in this graph, 29.5, as well as the midpoint, is also at 0. Even if I figured that out, I still have the problem of getting the first crest to be at 16, because according the problem, that's how long it takes between now and the next full moon.
It would be better to take the absolute value, that would make more sense. Let’s start with your function: $$y=\left| \sin\left( \frac{2\pi}{29.5}x\right)\right |$$ Taking the absolute value halves the period, so to maintain a period of $29.5$ days, divide by $2$:
$$y=\left| \sin\left( \frac{\pi}{29.5}x\right)\right|$$
Notice that we get a peak at $x=\frac{29.5}{2}= 14.75$, which is $1.25$ off from $16$. So, we have to shift the graph $1.25$ units to the right as follows:
$$y=\left| \sin\left( \frac{\pi}{29.5}(x-1.25)\right)\right |$$ This is it. Now to answer your question, $y(0)\approx 0.133 = 13.3\%$.