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$f(x) = 2-4\cos(2x-\frac{1}{3}\pi)$

How do you get this function from as standard function?

My answer:

$ y = \cos x $

↓ Multiply with the x-axis, -4

$y=-4\cos x$

↓ Multiply with the y-axis, 0.5

$y=-4\cos 2x $

↓ Translation $(\frac{1}{6}\pi, 2)$

$f(x) = 2-4\cos(2x-\frac{1}{3}\pi)$

Is this answer correct? The only part I am really concerned about is the final translation, with the $(\frac{1}{6}\pi)$ instead of $(\frac{1}{3}\pi)$.

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    I'm sorry guys, I was kind off in a bad mood when I asked it. I've got a big test coming up tomorrow and I missed all lessons because of an illness so I basically have to use you guys as my teachers. Also, 'multiply with the x-asis', in my language it is called that, I thought (like often) that I could just directly translate it into english and it would still be understandable, guess not..2012-09-17

1 Answers 1

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$\frac\pi6$ seems to be correct:

You have to consider first, what is happening in 'machine' $\cos$ with your variable $x$. Those that happen before applying $\cos$ will take action horizontally and inverted.

  • First plot $\cos x$,
  • then $\cos(x-\frac\pi6)$ -- [shift to the right by $\frac\pi6$]
  • then $\cos(2x-\frac\pi3)$ -- [shrink horizontally by $1/2$]
  • then $-4\cos(2x-\frac\pi3)$ -- [reflect along the x-axis because of the $-$ and multiply vertically by 4]
  • finally add 2 -- [vertically]