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Is there any type of function that when graphed would show a curve which intersects the x axis multiple times, with each point being an arbitrary distance from the last?

I mean, not like a trig function where each intersect is the same distance from the last. (2,0); (4,0); (6,0); (8,0). And not like a spiral where the distance gets bigger and bigger (or smaller) (2,0); (4,0); (8,0); (16,0);

But for example, some curve which intersects x-axis at (2,0); (6,0); (14,0); (15,0); (20,0); (122,0)...

Does that type function exist?

If so, is it possible to solve/get the equation, given only those intersect points?

I wouldn't need the exact equation of any particular curve. Just the equation of any curve that happens to intersect x-axis at whatever given arbitrary x values. Is that at least that possible to do?

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2 Answers 2

If you specify an arbitrary finite set of points $\{a_1, a_2, a_3, \dots, a_n\}$, the polynomial function $$f(x)=(x-a_1)(x-a_2)(x-a_3) \cdots (x-a_n)$$ has the desired property.

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If I'm interpreting your question correctly, you want to know if, given an increasing sequence of points $(a_n)$ in $\mathbb R$, we can define a continuous function $f:\mathbb R\to\mathbb R$ such that $f(x)=0$ if and only if $x=a_i$ for some $i$ (that is, the graph of $f$ intersects the $x$ axis precisely at the points $(a_i,0)$). The answer is yes, in fact there are infinitely many curves which will suffice. They might be hard to describe however. The simplest example I can think of is $$f(x)=(-1)^i\sin\left(\frac{(x-a_i)\pi}{a_{i+1}-a_i}\right)\text{ for } a_i\leq x\leq a_{i+1}$$ and $f(x)=0$ otherwise. This is continuous but not differentiable, and has the advantage of actually crossing the $x$-axis at each $a_i$ rather than just touching it.

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Ok cool. It sounds like you know what you're talking about. Let me try to explain differently. It doesnt matter if it's crossing or just touching the x-axis. It doesn't matter if it hits the points in any particular order - the graph could look like a curlicue... It just has to satisfy one condition: that (for example) when y=0; x= 2,6,14,15,20, and 122. (or 14,6,122,2, and 20 - or any order is fine, as long as it hits x at exactly these points when x is between the given max and min). –  monkey blot Jun 16 '12 at 6:17
    
Will it be possible to find a function where I'm not just tacking on another term to explain each point - but just a brief equation that describes a curve, not the points... (but has those points). Any other points on the curve do not matter, there could even be more y=0 points, as long as their corresponding x vals are outside the range of the given points. Whew! Sorry this question is ridiculously tricky. –  monkey blot Jun 16 '12 at 6:18

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