Amplitude, Period, and Displacement?

**Smokinoakum** · March 19th 2011, 07:39 PM

Hi all...after finishing my homework tonight I decided to keep going in the chapter to get ahead. Yeah, I was also enjoying myself too!

The next part of the chapter deals with Amplitude, Period, and Displacement.

Here is a sample question from the book.

Determine the amplitude, period, and displacement of the given function. Then sketch the graph of the function.

$\displaystyle y = 2\ cos \displaystyle \left(x\, + \frac{\pi}{12}\right)$

Could someone break this type of problem down to it's parts so I can work my way through this section of the chapter?

Once again, I'm taking my math online which means I've basicly become my own teacher.

**Oiler** · March 19th 2011, 07:54 PM

Hey,
you could plot the function against the graph of $y = cosx$ , Translate the graph in stages, i.e change the amplitute $y = 2cosx$ and observe the result. Then you can look at the graph of $y = cos(x+\frac{\pi}{12})$ (Phase shift of $\frac{\pi}{12}$ .), how to draw $y = 2cos(x+\frac{\pi}{12})$ should become obvious.

**Smokinoakum** · March 19th 2011, 07:58 PM

Originally Posted by Oiler

Hey,
you could plot the function against the graph of $y = cosx$ , Translate the graph in stages, i.e change the amplitute $y = 2cosx$ and observe the result. Then you can look at the graph of $y = cos(x+\frac{\pi}{12})$ , how to draw $y = 2cos(x+\frac{\pi}{12})$ should become obvious.

Thanks for responding Oiler. I think first I need to understand the proper way to find the amplitude, period, and displacement of the given function. After that I know I could use the calculator to see the graph.

Does that make sense?

**Oiler** · March 19th 2011, 08:08 PM

Given, $f(x) = A sin(Bx-C) + D$
The Amplitude is $A$
The vertical shift is $D$
The period will be given by $\frac{2\pi}{B}$
And you can work out the frequency by $f = \frac{1}{period} = \frac{1}{\frac{2\pi}{B}}=\frac{B}{2\pi}$
The phase shift is $\frac{C}{B}$
Hopefully that helps..

**Smokinoakum** · March 19th 2011, 08:11 PM

Originally Posted by Oiler

Given, $f(x) = A sin(Bx-C) + D$
The Amplitude is $A$
The vertical shift is $D$
The period will be given by $\frac{2\pi}{B}$
And you can work out the frequency by $f = \frac{1}{period} = \frac{1}{\frac{2\pi}{B}}=\frac{B}{2\pi}$
The phase shift is $\frac{C}{B}$
Hopefully that helps..

Yes that helps a lot. I have added that to my notes.

So after looking this over this is what I came up with.

Determine the amplitude, period, and displacement of the given function. Then sketch the graph of the function.

$\displaystyle y = 2\ cos \displaystyle \left(x\, + \frac{\pi}{12}\right)$

The Amplitude is = $2$

The period = $\frac{2\pi}{1}$ = $2\pi$

The displacement = $-\frac{\pi}{\frac{12}{1}}$ = $-\frac{\pi}{12}$

**Smokinoakum** · March 19th 2011, 10:30 PM

I've worked through about ten like the previous post and they were progressively getting harder and then a came upon this one...

Find the function and graph it for a function of the form $\displaystyle y = 7\ sin \displaystyle \left(7x\ + c)\right$ that passes through $\left(-\frac{\pi}{5}, 0)\right$ and for which $c$ has the smallest possible positive value.

This one has me a bit intimidated.

How do I go about this one?

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