Question

How do you find the period and asymptote of this graph? y= 2sec 4x

Answer 1

The period is how long it takes for the function to repeat itself. I usually find it helps to rewrite the equation like this.\[\large y = \frac{2}{\cos (4x)}\]Now, if the period is \(T\), then the function will be the exact same when we substitute \(x\) with \(x+T\). This inequality can be written in symbols, and then simplified as follows.\[\large y=\frac{2}{\cos(4x)}=\frac{2}{\cos(4(x+T))} \Rightarrow \cos (4x)=\cos(4x+4T)\]This should give you enough of a nudge to reason out what the value of \(T\) is.

Since asymptotes occur when we have a zero in the denominator, the asymptotes of \(y\) are simply going to occur when we have \(\cos(4x)=0\). I'll leave it to you to find those values.

Answer 2

Thanks!!