Melody

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.

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.

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. ]

Thanks Tiggsy :)

I cannot open this.

sin(2pi+pi/6)

this must equal    sin (pi/6)  Becasue it is just sine (pi/6 + 1 rotation)

sin(pi/6) = 0.5

I really do not know what this means "Note that identically sized groups are indistinguishable."

BUT

In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group?

The groups will be      MW,     MW,     MWW

First, How many ways can MWW be chosen     3*4*3 

Now how many ways can a MW be chosen for the next group.    2*2

The left overs make the last group.

So I have     3*4*3*2*2 = 144 ways

2.

The mean average temperature in Buffalo NY is 47.5 degrees. The temperature fluctuates 23.5 degrees above and below the mean temperature. If t =1 represents January, the phase shift of the sine function is 4. 

a. write a model for the average monthly temperature in Buffalo.

let x=month where December is 0 and 12 etc

let y= temperature

let a= amplitude

let h=average temperature

The wavelength will be 2pi/n

\(y=asin[n(x-k)]+h \)

h=47.5

a=23.5

k=4

The wavelength is 12 months and wavelength = 2pi/n

so

\(12=\frac{2\pi}{n}\\ n=\frac{2\pi}{12}=\frac{\pi}{6}\)

So we have:

\(y=asin[n(x-k)]+h\\ y=23.5sin[\frac{\pi}{6}(x-4)]+47.5\\ \)

b. according to the model, what is the average temperature in March?    x=3

23.5*sin(pi/6(3-4))+47.5 = 35.75

So, according to the modal the average March temp is 36 degrees

c. according to the model, what is the average temperature in August?     x=8

23.5*sin(pi/6(8-4))+47.5 = 67.851596988924

So, according to the modal the average August temp is   68 degrees

Here is the graph

1. A leaf floats on the water bobbing up and down. The distance between its highest and lowest point is 4 centimeters. It moves from its highest point down to its lowest point and back to its highest point every 10 seconds. Write a cosine function that models the movement of the leaf in relationship to the equilibrium point.

**I got y = 4 cos(π/5)m as my answer - where y is the movement of the leaf and m is the relationship of the leaf to the equilibrium point, but I'm not sure if it is right. 

\(y=acos(nt)\)

a is the ampitude and that is 2

It is 2 cm from the centre of the wave to the crest (or trough)

The angle is in radians. if n=1 the period is 2pi   

in general though the period is 2pi/n

The question requires that the period is 20     

\(20=\frac{2\pi}{n}\\ 20n=2\pi\\ 10n=\pi\\ n=\frac{\pi}{10}\)

\(y=2cos(\frac{\pi t}{10})\qquad \text{where t is in seconds}\\\)

LOL  MUST BE!