1. Suppose that the graph of a certain function, y=f(x), has the property that if it is shifted 20 units to the right, then the resulting graph is identical to the original graph of y=f(x). What is the smallest positive a such that if the graph of \(y=f\left(\frac x5\right)\) is shifted \(a\) units to the right, then we know that the resulting graph is identical to the original graph of \(y=f\left(\frac x5\right)\)?
I wasn't sure so I experimented, I decided to use a sine graph with a wavelength of 20
y=sin nx
the wavelength is 2pi/n = 20
so n = pi/10
so my experimenting graph is
\( y=sin(\frac{\pi }{10}x)\\ y=sin(\frac{\pi }{10}x)=sin(\frac{\pi }{10}(x-20))\\ let \;\;\frac{x}{5}\;\;be\;\;substituted\;\;for\;\;x\\ y=sin(\frac{\pi }{10}\frac{x}{5})=sin(\frac{\pi }{10}(\frac{x}{5}-20))=sin(\frac{\pi }{10}(\frac{x-100}{5})) \)
So the smallest a that we know with certainty is 100.
Here is the graph i was playing with.
https://www.desmos.com/calculator/oonutt20vz
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Let me see if I can do this more simply.
\( y=f(x)=f(x-20)\\ let \;\;\frac{x}{5}\;\;be\;\;substituted\;\;for\;\;x\\ y=f(\frac{x}{5})=f(\frac{x}{5}-20)=f(\frac{x-100}{5}) \)
There you go, that was simple. The right shift is 100.