Electrical – Arduino analog in with 12v

Most microcontroller ADCs have a pin for the positive ADC reference voltage, and some will also have one for the negative reference. Those that don't will use ground as the lower reference. Then the ADC's reading is \$\dfrac{V_{IN} - V_{REF-}}{V_{REF+} - V_{REF-}} \times Maximum\ reading\$. If Maximum Reading is 1023, then \$V_{IN} = V_{REF-}\$ will give you 0, and \$V_{IN} = V_{REF+}\$ will result in a 1023 reading.

As for the unexpected values you see, check this answer.

All you need to do is to scale the equation based on the new units. If X was previously Volts and now is in units of 5 V / 1023. So scale your coefficients by 5 / 1023 to get the same Y with the new X units.

While you're at it, you probably want to scale the coefficients so that Y is in something more useful than percent. Percent may be appropriate for displaying a fraction to humans but is pretty useless and inconvenient otherwise. Either make it a fraction from 0 to 1 so you can easily multiply it by whatever maximum PWM value your hardware needs, or incorporate this into the equation in the first place. So start by scaling the coefficients down by 100, then up by the maximum hardware duty cycle value. After scaling to the new X as above, you have the equation from A/D reading to PWM output value.

That answers your question, but none of this is what you realy want. It seems you want to implement a control system. For that it is good to know that the control output (the PWM duty cycle value) produces a roughly linear system output (the ball position). The feedback gains you use then are a function of the system dynamics. Your X coefficient represents open loop gain which will effect the feedback coefficients, but it's probably simpler to find the right values by imperical tweaking.

A simple control scheme for which there is much information out there is called "PID". That stands for Proportional, Integral, and Derivative. A PID controller is a whole subject onto itself, but there is much available out that out there so there is little point repeating it here.