# Thread: graph of equation over given interval

1. ## graph of equation over given interval

Identify the graph of the equation over the given interval.

y=3sin (2x-pi/2)

x is greater than or equal to -pi/4 and x is less than or equal to 3pi/2.

I think I found the period to be: pi
the amplitude is 3.
the phase shift is pi as well?

I'm unsure as to what to do after this though.

My choices are these, but I don't know how to determine which is correct. I'm not just looking for an easy answer, I really want to try to understand this. Thanks so much guys.

2. Actually, phase shift is $\frac{\pi}{4}$ since: $y = 3\sin \left(2x - \frac{\pi}{2}\right) = y = 3\sin \left[ 2\left(x - {\color{red}\frac{\pi}{4}}\right)\right]$

Now from your typical $y = \sin x$ graph, make the changes accordingly. Change its amplitude to 3. Shift it to the right $\frac{\pi}{4}$ units. Adjust the period.
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Alternatively ...

To make things easier, note that: $\sin \left(\theta - \frac{\pi}{2}\right) = -\cos \theta$

So we have: $y = 3\sin \left(2x - \frac{\pi}{2}\right) \ \Leftrightarrow \ y = 3 (-cos 2x) = -3 \cos (2x)$

Much easier to graph