How do you graph and list the amplitude, period, phase shift for y=1/2tantheta?

Feb 25, 2018

The amplitude is $\frac{1}{2}$, the period is $\frac{\pi}{1} = \pi$, and the phase shift is $0.$

Explanation:

Let's compare our equation to the following:

$y = A \tan \left(b \theta + c\right) + d$

From this equation, we can deduce the following:

$A$ is the amplitude.

The period can be calculated from the formula $\frac{\pi}{b}$.

The phase shift is $c$ units to the left if $c > 0$ and $c$ units to the right if $c < 0$ and no shift at all if $c = 0$.

The vertical shift is $d$ units up if $d > 0$, $d$ units down if $d < 0$, and no vertical shift if $d = 0.$

For $y = \frac{1}{2} \tan \left(\theta\right)$, we see that $A = \frac{1}{2} , b = 1 , c = 0 , d = 0$

Therefore, the amplitude is $\frac{1}{2}$, the period is $\frac{\pi}{1} = \pi$, and the phase shift is $0.$

Since we have no phase shift, we have vertical asymptotes at $x = \frac{n \pi}{2}$ where $n$ is any integer.

$\tan \left(0\right) = 0$, so we have the point $\left(0 , 0\right)$ on our graph.

As we approach $x = \frac{\pi}{2}$, $\frac{1}{2} \tan \left(\theta\right)$ begins to increase faster and faster, running nearly parallel to $x = \frac{\pi}{2}$ but never touching it.

As we approach $x = - \frac{\pi}{2}$, $\frac{1}{2} \tan \left(\theta\right)$ begins to decrease faster and faster, running nearly parallel to $x = - \frac{\pi}{2}$ but never touching it.

Due to our amplitude of $\frac{1}{2}$, the graph becomes slightly vertically compressed compared to the graph of $y = \tan \left(\theta\right)$ (but the asymptotes remain unchanged).

We have just graphed a full period of $y = \frac{1}{2} \tan \left(\theta\right)$. This pattern repeats on intervals of $\pi$, with the $x$-intercepts of the graph being at $\left(\pi n , 0\right)$ where $n$ is any integer.

For comparison, the graphs of $y = \tan \left(\theta\right)$ and $y = \frac{1}{2} \tan \left(\theta\right)$ (respectively) so we can see the horizontal compression and periodic behavior:

graph{y=tan(x) [-10, 10, -5, 5]}

graph{y=1/2tan(x) [-10, 10, -5, 5]}