Finding the intersection on a curved line (graph)

I'm a math novice and not familiar with the jargon, though I've made my best effort to look up terms and hopefully I picked the right subforum. If not I hope a mod will move it to the subforum it belongs to, rather than punish me. It seems you're expected to be an actual student of the field to participate in this forum given how the rules are laid out...

Anyway, here's my problem stated as best I can. If I don't use the right words or forms of expression please ask me to clarify further. I know what I mean even if I don't say it right the first time, hopefully people can read between the lines.

I have a 2d graph with an X and Y axis. There are 20 parts/points/subdivisions(?) on the X axis, where the value of X is the same as the number of that point. at each of the 20 integer points on the X axis, Y(X) (<- by that I mean: the value of Y at point X...?) is Y(X+1)/ 3 x 2. in other words Y(X) is Y(X-1)*1.5, or for each +1 increment to X starting at a whole number, the value Y associated with X increases by 50%.

Y(20) is 999999 in the problem i'm trying to solve, if that matters.

What I'd like to know is, if an imaginary curved line ran through the nodes that this equation would plot on a graph (the nodes being at each of the 20 points on the X axis), how would I go about finding the value of Y on a fraction of X, for instance what is Y where X is 17.35 ? I'm imagining something along the line of gradients inside of gradients... to get from Y(18) to Y(19) you have to increase Y(18) by 50% but what order of increase to get from Y(18.25) to Y(18.75) ?

Hopefully people will recognise the level i'm at and i'll evade bamboozlement when discussing the answer to this problem. Or if i'm simply in the wrong place in my quest for an answer perhaps someone can orientate me to the right place. Thanks in advance for any help offered.

Re: Finding the intersection on a curved line (graph)

Quote:

Originally Posted by

**caruga** at each of the 20 integer points on the X axis, Y(X) (<- by that I mean: the value of Y at point X...?) is Y(X+1)/ 3 x 2.

Do you mean Y(X**-**1)/ 3 x 2?

There are many smooth functions that pass through these 20 points, but the simplest probably is $\displaystyle y(x) = A(1.5)^x=Ae^{x\ln 1.5}$ for some constant A. Here $\displaystyle e^x$ is the exponential function and $\displaystyle \ln x$ is the natural logarithm function. Indeed, then $\displaystyle y(x+1)=A(1.5)^{x+1}=A(1.5)^x\cdot1.5=1.5y(x)$.

To find A, use the fact that $\displaystyle y(20)=A(1.5)^{20}=999999$, from where $\displaystyle A=999999/(1.5)^{20}$.

Re: Finding the intersection on a curved line (graph)

Quote:

Originally Posted by

**emakarov** Do you mean Y(X**-**1)/ 3 x 2?

Nope, that would mean that Y decreases as X increases. I was giving the descending version: 666666 = (999999 / 3 *2)

Quote:

There are many smooth functions that pass through these 20 points, but the simplest probably is $\displaystyle y(x) = A(1.5)^x=Ae^{x\ln 1.5}$ for some constant A. Here $\displaystyle e^x$ is the exponential function and $\displaystyle \ln x$ is the natural logarithm function. Indeed, then $\displaystyle y(x+1)=A(1.5)^{x+1}=A(1.5)^x\cdot1.5=1.5y(x)$.

To find A, use the fact that $\displaystyle y(20)=A(1.5)^{20}=999999$, from where $\displaystyle A=999999/(1.5)^{20}$.

I'll try wrap my mind around what you said. :) Thanks for the input.