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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.)

Mar 27, 2018

#1
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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. ]

Mar 27, 2018
#2
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You rewrote them both correctly

Mar 27, 2018
edited by Guest  Mar 27, 2018
edited by Guest  Mar 27, 2018
#3
+100025
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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.

Melody  Mar 27, 2018
edited by Guest  Mar 27, 2018
#4
+100025
+1

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

-------------------

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.

Mar 27, 2018
#5
+100025
+1

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:

Mar 27, 2018