How do I find the period of the sine function $y = 20\sin\left[\frac{5 \pi}{2}\left(\frac{x -2}{5}\right)\right] + 100$

Question

Using Desmos I can see the period is $0.8$ but how do I get there? I understand that the period is $2\pi/$co-efficient of $x$ but the $-2/5$ is throwing me off.

Answer 1

Shifts are to be neglected, hence, if $T$ is a period, we have $$ \frac{5\pi}{2}T=2\pi, $$ which gives $T=0.8$.

Answer 2

Transformations of trigonometric functions, here is Sine in particular, follow this schema. $$A\sin(B(x-C))+D$$
As mentioned above only one of these four values affects the period. C will shift the graph horizontally while D will shift the graph vertically. A will stretch or shrink the amplitude, so that leaves B as our "period-morphing" parameter.

In the comments above see that $P=\frac{2\pi}{B}$. So all you need to do is substitute the value of B in to solve for the period.

$$P=\frac{2\pi}{\frac{5\pi}2}=\frac{2\pi*2}{5\pi}=\frac45$$

Keep in mind
If there were not double parentheses in this problem you would have to factor out the B parameter from the potential binomial inside as best practice. I assume that you are a high school or 1st year college student so you would likely be asked more than one follow-up question to this function.