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Edit: Thanks to those commenting, I fixed my issue and now have reached the below function, as I had aimed. My remaining question would be how I might describe this function. I have taken the function from (Function For Sine Wave Between Two Exponential Cuves) where they were tackling a similar problem, and I want to be able to describe what the function is doing, but I don't know if it is causing oscillation or using midpoints between the linear functions to inform the sine wave.

I've attached an image of the new function: New function

end edit

I'm trying to make a sinusoidal function that dilates as it progresses between two linear functions so that its peaks and troughs are always in contact with the linear functions. To do this I have worked with the function;$$\frac{y_{3}+y_{2}}{2}-\frac{y_{3}-y_{2}}{2}\sin x $$ where y_3 and y_2 are linear functions built on a series of points and I got this as a result: Multi-sinusoidal function?

I've called it a 'multi'-sinusoidal as it looks like the function has made varying sine waves, each that has a peak for one of my table data points. For example, the largest sine wave matches my greatest point's y value; 6.3, and there is also a sine wave for the smallest value; 4.8.

Is there any way I could merge these functions so that it is just one sine wave, or there other mathematical 'boundary' formulas that I could use to tell the sine wave to equal the value of the linear function point it corresponds to?

Thanks !!

cadasc_56
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  • If there is just one $y_3$ function, and just one $y_2$ function, and they are each one of the blue lines on your graph, I don't understand why there are multiple green sine waves. – JonathanZ Feb 24 '24 at 07:21
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    You've created a different wave function for each data point, haven't you? That's not what you've described. $y_3$ should be a function of $x$, and so should $y_2$. It looks like you've graphed the lines corresponding to them, but you haven't figured out their slope & intercept and used those to generate your sine wave. Can I ask what software you are using to generate your image? – JonathanZ Feb 24 '24 at 07:29
  • I've used desmos. That's a good idea though, I will try it and edit the post if it has changed anything. For context the y3 and y2 functions are approximate functions modelled off data points, so your comment is probably right in reasoning why it has made the multiple functions. – cadasc_56 Feb 27 '24 at 03:40
  • That's nicer! As for a description, I would describe it as "a decaying sine wave oscillating within an envelope bounded by two (slowly) converging lines". But that's pretty general, and you should be sure to include any other features that are relevant to your particular requirements. – JonathanZ Feb 27 '24 at 04:57
  • Obviously $y_2$ and $y_3$ are numbers and not functions in your last example. – user Feb 27 '24 at 12:08

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