Question

Period of the function $f\left(x\right)=4\sec (3x-\frac{\pi }{4})$ is

• A. $\frac{4\pi }{3}$
• B. $2\pi$
• C. $\frac{3\pi }{4}$
• D. $\frac{2\pi }{3}$

Answer 1

The correct option is D $\frac{2\pi }{3}$

Let us first draw the graph $y=4\sec (3x-\frac{\pi }{4})$.
The fundamental function involved is $\sec x$ whose period is $2\pi$ and graph is given by


Apply a stretch by 3 units to the graph of $\sec x$ to get $y=\sec 3x$ as shown below

We can see that period of $y=\sec 3x$ is $\frac{2\pi }{3}$.
or we can also use the technique, if T is period of $f\left(x\right)$ then period of $f\left(ax\right)$ is $\frac{T}{|a|}$.
Now, apply the horizontal shift to above graph, to get $y=\sec (3x-\frac{\pi }{4})$ as shown below


Here, we can see horizontal shift doesnot effect the period and period remains same as $\frac{2\pi }{3}$.
Now stretch the above graph by 4 units to get the graph of $y=4\sec (3x-\frac{\pi }{4})$ as shown below

Again we can see that vertical stretch doesnot effect the period and period remains same as $\frac{2\pi }{3}$.