Hi Annaoh!
where does the "base" function start ... I understand the transformations - dilate by 2, extend period by 2 (I think) and the vertical shift of 1, but then get confused over the -pi / 3 ?
When the question says "standard" cosine function it is referring to \(y=\cos \left(x\right)\) and it is asking - what transformations need to be applied to \(y=\cos \left(x\right)\) to get \(f\left(x\right)=2\cos \left(2x-\frac{\pi }{3}\right)+1\).
It might be easier to look at \(f\left(x\right)\) as \(y=2\cos \left(2\left(x-\frac{\pi }{6}\right)\right)+1\) and determine the transformations by inspection.
If you struggle to see the transformation like that you can alternatively rearrange and solve the following to figure out what the transformation is as \(\left(x,\:y\right)\rightarrow \left(x',\:y'\right)\) where \(y=\cos \left(x\right)\) and \(y'=2\cos \left(2\left(x'-\frac{\pi }{6}\right)\right)+1\)
Rearranging gets us \(y'=2\cos \left(2\left(x'-\frac{\pi }{6}\right)\right)+1\Rightarrow \:\frac{y'-1}{2}=\cos \left(2\left(x'-\frac{\pi }{6}\right)\right)\)
Now we solve \(\frac{y'-1}{2}=y\) and \(2\left(x'-\frac{\pi }{6}\right)=x\) for \(y'\) and \(x'\). Solving those equations will get us \(y'=2y+1\) and \(x'=\frac{x}{2}+\frac{\pi }{6}\)
Therefore the mapping of the transformation: \(\left(x,\:y\right)\rightarrow \left(x',\:y'\right)\) (from here we just sub in \(x'\) and \(y'\)) is \(\left(x,\:y\right)\rightarrow \left(\frac{x}{2}+\frac{\pi \:}{6},\:2y+1\right)\)
This represents the transformations that were applied to \(y=\cos \left(x\right)\) to get \(f(x)\), which are:
- Dilation of factor 2 from the \(x\) axis
- Dilation of factor \(\frac{1}{2}\) from the \(y\) axis
- Translation of \(\frac{\pi }{6}\) units in the positive direction of the \(x\) axis
- Translation of 1 unit in the positive direction of the \(y\) axis - (which you correctly said)
do I need to do the transformations in some sort of order?
As mentioned by Billuminati, the order of transformations does matter - if you for instance translated and the translation was followed by a dilation you would need to also dilate the translation if you know what I mean. So as a general rule of thumb dilations first when determining the sequence of transformations.
plus I dont have a calculator to help me figure this out.
I think this question assumes you have memorised the important values of the unit circle below