Question

Hello this is alegra 2 maybe geometry. I need help on sketching once cycle of the graph on each function

Answer 1

In this session, we will focus on plotting the function

$y=2sin(6\theta)$

and also finding its amplitude and its period.

To do so, we will use the following:

- the sine function has a period of 2pi.

Given the function of the form

$Asin(B\theta)$

its amplitude would be A and its period would be given by the expression

$\frac{2\pi}{B}$

so if we compare this general expression to the function we are given, we can see right away that A=2 and that B=6, since

$Asin(B\theta)=2sin(6\theta)$

this means that the amplitude of the given function is 2 and the period would be

$\frac{2\pi}{6}=\frac{\pi}{3}$

(the period would be pi/3)

To plot this function, we will first recall how the plot of sin(theta) looks like.

From the picture, we can see that there are 4 relevant angles: 0, pi/2, pi, 3pi/2 and 2pi. So the graph of this new function would be exactly the same. The only thing that changes is the scale on the x axis. We will first find the equivalent of this 4 points in the new scale. So we want to solve the following equations

$\begin{gathered} 6\theta=0 \\ 6\theta=\frac{\pi}{2} \\ 6\theta=\pi \\ 6\theta=2\pi \end{gathered}$

We can find the values by simply dividing both sides by 6. So we get

$\begin{gathered} \theta=\frac{0}{6}=0 \\ \theta=\frac{\pi}{2\cdot6}=\frac{\pi}{12} \\ \theta=\frac{\pi}{6} \\ \theta=\frac{3\pi}{2\cdot6}=\frac{\pi}{4} \\ \theta=\frac{2\pi}{6}=\frac{\pi}{3} \end{gathered}$

Also, note that the amplitude is 2, so instead of going up until 1 (or down until -1) we go until 2 (or -2). So the new graph looks like this