What is the period of the function three tan multiplied by two 𝑥 minus 𝜋 over five?
To help us solve this problem, I’m actually drawing a graph. And this graph is 𝑦 equals tan 𝑥. And you can see here this function as it’s shown. What we can notice is that the function is actually repeating. And it’s this repeating nature of the function that’s 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 function’s cycle. And I’m actually showing that one on our graph because what I’ve shown is the length of our function’s 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 𝜋 over two to 𝜋 over two, we could see that actually the function’s being repeated then. Or if we go from zero to 𝜋, we can also see that again. We’ve got the function repeated cause it’s the same part of our function. So therefore, we can say that the period of our function is 𝜋 because we could see that both repeats took place in a space of 𝜋.
Okay, great. So we now know that the function tan 𝑥 has a period of 𝜋. Okay, so shown is how to find the period. But also, it’s gonna be useful because we’ll 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 we’re going to rearrange it into is this form, which is 𝑎 tan and then 𝑏𝑥 minus 𝑐 plus 𝑑. And if we do that, we get that the function is equal to three tan and then we have two over five 𝑥 minus 𝜋 over five. We don’t have to worry about the plus 𝑑 because we didn’t 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 𝜋 over the absolute value of 𝑏. So we have 𝜋. And that’s 𝜋 in our formula because we’ve got the period of the function 𝑦 equals tan 𝑥 which we showed earlier. And then on the denominator, we have 𝑏. Well, it’s the absolute value of 𝑏. We get 𝑏 from the form that we’ve rearranged our function into. And it’s the absolute value because we’re only interested in the positive values.
So therefore, if we substitute in our values, we get the period of our function is equal to 𝜋 divided by the absolute value of two over five. And the reason we get that is because that’s our 𝑏 when we’ve rearranged our function. Okay. So we can do 𝜋 divided by two over five. And we don’t have to worry about the absolute value because it’s already a positive value. Well, this is gonna be equal to 𝜋 multiplied by five over two. Cause you’ll see if we divide by a fraction, then we actually multiply by the reciprocal. So we got 𝜋 multiplied by five over two.
So therefore, we can say that the period of our function three tan multiplied by two 𝑥 minus 𝜋 over five is gonna be equal to five 𝜋 over two.