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

`(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

`(2\pi)/(6)=(\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=(\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=(0)/(6)=0 \\ \theta=(\pi)/(2\cdot6)=(\pi)/(12) \\ \theta=(\pi)/(6) \\ \theta=(3\pi)/(2\cdot6)=(\pi)/(4) \\ \theta=(2\pi)/(6)=(\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