Measure time for a change in voltage

So I'm working on a project that requires me to know the curve of voltage changing from X to 0 or vise versa on a pull up/down resistor. So basically be able to measure the voltage as it is changing from X to 0, to be able to plot the curve. I know that the Uno's analog sample rate is 10,000 /s. Does anyone know of projects that do something similar or if it's possible at that sample rate?

This sounds like the sort of question where a test with a simple program would give you answers very quickly. Or would give you more specific questions.

Do you have an oscilloscope so that you can verify the Arduino results?

...R