### Video Transcript

What is the period of the function three tan multiplied by two ๐ฅ minus ๐ over five?

To help us solve this problem, Iโm actually drawing a graph. And this graph is ๐ฆ equals tan ๐ฅ. And you can see here this function as itโs shown. What we can notice is that the function is actually repeating. And itโs this repeating nature of the function thatโs gonna help us with the first part of the question. Because first of all, we want to know what is a period.

Well, the period is the length of a functionโs cycle. And Iโm actually showing that one on our graph because what Iโve shown is the length of our functionโs cycle. Because you can see that within the cycle, the function has actually repeated itself. So we could see that this one here, if we went from negative ๐ over two to ๐ over two, we could see that actually the functionโs being repeated then. Or if we go from zero to ๐, we can also see that again. Weโve got the function repeated cause itโs the same part of our function. So therefore, we can say that the period of our function is ๐ because we could see that both repeats took place in a space of ๐.

Okay, great. So we now know that the function tan ๐ฅ has a period of ๐. Okay, so shown is how to find the period. But also, itโs gonna be useful because weโll use that later on. To be able to find the period of our function, first of all, what we want to do is we actually want to rearrange it into a different form. And the form that weโre going to rearrange it into is this form, which is ๐ tan and then ๐๐ฅ minus ๐ plus ๐. And if we do that, we get that the function is equal to three tan and then we have two over five ๐ฅ minus ๐ over five. We donโt have to worry about the plus ๐ because we didnโt have anything in the original function.

Okay, great. But why do we do this? Well, we do this because actually it means we can use this formula which tells us the period of our function is equal to ๐ over the absolute value of ๐. So we have ๐. And thatโs ๐ in our formula because weโve got the period of the function ๐ฆ equals tan ๐ฅ which we showed earlier. And then on the denominator, we have ๐. Well, itโs the absolute value of ๐. We get ๐ from the form that weโve rearranged our function into. And itโs the absolute value because weโre only interested in the positive values.

So therefore, if we substitute in our values, we get the period of our function is equal to ๐ divided by the absolute value of two over five. And the reason we get that is because thatโs our ๐ when weโve rearranged our function. Okay. So we can do ๐ divided by two over five. And we donโt have to worry about the absolute value because itโs already a positive value. Well, this is gonna be equal to ๐ multiplied by five over two. Cause youโll see if we divide by a fraction, then we actually multiply by the reciprocal. So we got ๐ multiplied by five over two.

So therefore, we can say that the period of our function three tan multiplied by two ๐ฅ minus ๐ over five is gonna be equal to five ๐ over two.