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Hi Melody, thank you so much for all your help!! You have really helped me out a lot! 

Hi NinjaDragons98,

Here are the graphs.  I have also included a colour coded line at the centre of the temperature limits.

Red         Los Angeles

Green     Osaka

Blue        Buenos Aires

https://www.desmos.com/calculator/d2xfeenjdv

23)  Osaka (B green) has the biggest temperature variation becasue at 20.71 degrees it has the biggest amplitude.

24) You can see from the graph that Osaka has the coldest temperatures so it is most likely to snow in winter.

ALSO

The lowest sin(anything) is is -1

so the lowest temperatures are 

              Los Angeles   = -8.086+76.08 = 68 degrees

              Osaka            = -20.71+68.97 = 48 degrees

              Buenos Aires = -13.03+72.38 = 59 degrees

so you can see from the algebra that Osaka gets the coldest.

25)  Explain why the function for Buenos Aires has a positive horizonatal shift as opposed to the negative horizontal shift for Osaka and Los Angeles.

Los Angeles and Osaka are in the Northern Hemisphere and Buenos Aires is int the Southern hemisphere.

Los Angeles     y=8.086sin(0.01863t-2.507)+76.08

Osaka              y=20.71sin(0.0168t-1.945)+68.97

Buenos Aires   y=13.03sin(0.01710t+1.331)+72.38

If there was no shift - like the black graph then The hottest day would be about 1st April.

The southern hemisphere had its hottest day earlier than that.  The positive shift  number actually moves the graph in the negative direction.

The northern hemisphere has its hottest day elater than that.  The negative shift  number actually moves the graph in the positive direction.

Here is another graph.  I have superimposed a sine wave that has no horizontal shift. It is the black one.

To make it easy to compare I have made the centre line comparable to the others and the period is one year. 

Incidental to make the period a year I made the coefficient of t equal to 2pi/364=0.01726

26)  When will Los Angeles (red) and Osaka (greeen) have the same daily high temp.

From the graph,

The red and green graphs intersect at about (138,77) and (264,81)

138/364*12 = 4.5494505494505495   That is the middle of April  Day 138

264/364*12 = 8.7032967032967033   That is the middle of  August,  Day 264

27)  Logic tells me that they have more chance of all being the same temp in the middle of Spring or Autumn.

So, from the choices,  that would be April 25th.

From the graphs I can see that they have the most chance of being the same between day 107 and day 138

Both these days are in April so again it will be April 25th.

28)  The vertical shift from 0 degrees for Osaka is 68 degrees (this will be the middle of the temperaure range).

This is less than the other 2 so overall Osaka will be the coldest.

This is answered in probably more detail than you needed NinjaDragon but hopefully you will be able to make sense of it.

:)

Feel free to ask question.

Ok NinjaDragon98 has got more questions...  It is always good to ask questions when you don't totally understand!    

There are 2 common ways to measure angles.  

One is in degrees where there are 360 degrees in a revolution

and the other in radians were there are 2pi radians in a revolution.

Yes it does make a difference what your graph trig functions in.

Say you are graphing sin(theta)

If it is in degrees then the wave length will be 360 (degrees)  

If it is in radians then the wave length will be  2pi  (radians)  = about 6.28

Of course   360 degrees = 2pi radans but the numbers on the graph will be in accordance with the units you are using.

All the graphs I have presented are in radians.When you are in Desmos, there is a little spanner on the right side.  If you click that you will see radians and degrees buttons at the bottome.  I have set my graphs to radians.

Whenever I talk about degrees in this question it is concerning temperature - NOT anglular measure.

Your logic for Q23 is fine.  The question suggests tht you draw the graphs and so if you answer from what you can see that is should be all that is required.

However the number in front of the sine function gives the amplitude of the wave.  The amplitude is the maximum distance from the middle.  Since 20.71 is the biggest amplitude Osaka will have the biggest temp variation.  It would be good for you to assimilate this into your understanding of sine and cosine graphs. :)

Q24. Yes your logic is fine. It is the same as mine. :)

I think that covers what you ahve asked so far...  Any more questions, yes? then keep asking.   

I do prefer you to ask your questions on the froum (on this thread if it is for this question) Then you can send me a note with the address stating that you would like more help :) 

If you do not message me I may not know that you have asked a question but I still would like that question to be asked on the forum :))

Thank you so much for all your help Melody!!

How do I do this?

Hi Melody, I too dont have a graphing calculator. What am I to do?