How do I tell if a graph is quadratic or exponential

[granite tundra]

So I am trying to tell if these points I have are quadratic or exponential here are some of the points

(1, 1) (2, 1) (3, 2) (4, 3) (5, 5)

I also will attach a graph of these points

Bonus points to anyone who can tell where these points came from (I think someone can)

[modern cave]

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[lone mantle]

Well, suppose it's an exponential function. That corresponds to a geometric sequence, right? Which corresponds to a constant common ratio.

Except I suppose you technically also get that in the case of a quadratic.

In this case, however, the first difference of the sequence is identical to the sequence.

In this specific context, you're looking at the Fibonacci sequence, which I know is exponential/geometric... kinda? But the proof for it isn't quite simple.

[granite tundra]

glad you noticed haha. I am trying to see if I can extend the defintion of the Fibonacci squence to the real numbers. at the moment I have noticed that if I stick only postive it is the exponential function (p^x)b where p is the golden ratio (I don't have phi copy and pasted atm) I do not know what b is and I may be incorrect fully on what the actual function I have right now. I have tried polynomial interpolation to see if I can find a pattern in it and create something I call interpolation series which is just an infinite polynomial you get from doing polynomial interpolation an infinite number of times. I did it up to 5 variables and noticed no patterns. I may return to it later on and check my old notes on the interpolation stuff to see if I have anything about the common patterns that show up in specfic functions

if there already has been a formula to take the Fibonacci sequence to the reals do not inform me of it because I wish to discover it on my own

[lone mantle]

Can I tell you the method?

[granite tundra]

is it possible for us to have a conversation that leads me to the method myself that way I am still thinking through it all

[lone mantle]

I mean, when I say "the method", I mean, like, the mathematical concept used.

I'm not sure if you'd have heard of it, though I'm more confident you could understand it once it was explained to you now.

[lone mantle]

I think I've got a reasonably good hint to give now, if you want it. It doesn't even name the concept, it just points you at it.

[granite tundra]

alrighty that works!

[lone mantle]

So you're already thinking about polynomials. What if, instead of a polynomial that produced the Fibonacci numbers as its output, you had one that had the Fibonacci numbers as its coefficients?

[granite tundra]

but it wouldn't still produce the Fibonacci numbers would it?

[lone mantle]

No, but if you gave this polynomial a name, such as $f(x) = \sum_{n = 0}^\infty f_nx^n$, you could try to manipulate it into a form that would yield a formula for $f_n$.

[median marsh]

Techie Literate

[granite tundra]

alrighty I will work from here! If I need any help may I dm/tag you