Question

identify the period and equation of one asymptote of the function ( f(x)=3\tanleft(\frac{3}{4}x
ight)-3 ).
period:
equation of one asymptote: ( x=)
answer attempt 1 out of 2
you must answer all questions above in order to submit.

Explanation:

Step1: Recall the formula for the period of \(y = A\tan(Bx - C)+D\)

The period of the tangent function \(y = A\tan(Bx - C)+D\) is given by \(T=\frac{\pi}{|B|}\). For the function \(f(x)=3\tan(\frac{3}{4}x)-3\), here \(B = \frac{3}{4}\).

$$T=\frac{\pi}{\frac{3}{4}}=\frac{4\pi}{3}$$

Step2: Recall the formula for the asymptotes of \(y = A\tan(Bx - C)+D\)

The asymptotes of the tangent function \(y = A\tan(Bx - C)+D\) are found by solving \(Bx - C=\frac{\pi}{2}+k\pi\), \(k\in\mathbb{Z}\). For \(f(x)=3\tan(\frac{3}{4}x)-3\) (where \(C = 0\)), we solve \(\frac{3}{4}x=\frac{\pi}{2}+k\pi\). Let \(k = 0\), then \(\frac{3}{4}x=\frac{\pi}{2}\), and \(x=\frac{2\pi}{3}\)

Answer:

Period: \(\frac{4\pi}{3}\)
Equation of one asymptote: \(x=\frac{2\pi}{3}\)