# How do you find the important points to graph y=3 tan (4x-pi/3)?

Jul 11, 2018

See explanation and graphs.

#### Explanation:

The period of $\tan \left(k x + c\right) = \frac{\pi}{k}$.

So, the period of $y = \frac{\pi}{4} = 0.7854$, nearly, and $\frac{\pi}{3} = 1.0472$,

nearly..

As $4 x - \frac{\pi}{3} \to \left(2 k + 1\right) \frac{\pi}{2} , y \to \pm \infty , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$.

Upon setting $k = - 1 \mathmr{and} 0$, one period about origin is

$x \in \left(- \frac{\pi}{24} , 5 \frac{\pi}{24}\right) = \left(- 0.1309 , 0.6545\right)$, nearly.

Direct graph for 10 periods, $x \in \left(0 , 7.854 0 3.927\right)$

(Slide the graph ($\leftarrow$) to see further periods)
graph{y - 3 tan ( 4 x - 1.0472 ) = 0[ 0 7.854 0 3.927] ]}

The inverse is $x = \frac{1}{4} \left(\arctan \left(\frac{y}{3}\right) + \frac{\pi}{3}\right)$

One-period graph using the inverse:
graph{x - 1/4(arctan( y/3 ) + 1.0472) = 0}