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 $y=f\left(\frac x5\right)$?
2. What are the coordinates of the points where the graphs of $f(x)=x^3-x^2+x+1$ and $g(x)=x^3+x^2+x-1$ intersect? Give your answer as a list of points separated by semicolons, with the points ordered such that their $x$-coordinates are in increasing order. (So "(1,-3); (2,3); (5,-7)" - without the quotes - is a valid answer format.)
+118609 I can't even read that question. I will rewrite it.
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 \(y=f\left(\frac x5\right)?\)
2. What are the coordinates of the points where the graphs of
\(f(x)=x^3-x^2+x+1\) and \(g(x)=x^3+x^2+x-1\) intersect?
Give your answer as a list of points separated by semicolons, with the points ordered such that their \(x\)-coordinates are in increasing order.
[So (1,-3); (2,3); (5,-7) is a valid answer format. ]
You rewrote them both correctly
+118609 I know, I put your expressions into the forum latex function.
If you do not know already how to do that you should learn.
You just open the latex box ( which is on the ribon) and copy your latex into it and then hit ok.
Just delete the dollar signs that surounds the code.
+118609 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.
+118609 2. What are the coordinates of the points where the graphs of $f(x)=x^3-x^2+x+1$ and $g(x)=x^3+x^2+x-1$ intersect? Give your answer as a list of points separated by semicolons, with the points ordered such that their $x$-coordinates are in increasing order. (So "(1,-3); (2,3); (5,-7)" - without the quotes - is a valid answer format.)
\(x^3-x^2+x+1 = x^3+x^2+x-1\\ -x^2+1 = x^2-1\\ 2=2x^2\\ x^2=1\\ x=\pm1\\ f(1)=1-1+1+1=2\\ f(-1)=-1-1-1+1=-2\\ \)
so the points of intersection are (-1,-2); (1,2)
check:
