First you need to decide whether time is a function of distance or distance is a function of time. That is, do you want to see a picture of how time changes as distance changes, or how distance changes as time changes? Typically, it is the second way.
When you say something is a function of something else, then the 'something else' is the variable on the horizontal axis. So, given that you want distance as a function of time, then time would be measured on the horizontal (what we normally call the x) axis. Distance would then be measured on the vertical or y axis.
The distance equals rate times time formula with which we are all so familiar is nothing more than an equation in slope-intercept form. Usually we don't have an 'initial distance' so the y-intercept is zero. But if there were an initial distance specified, that is a place other than 0 that you would be when time starts, it would be the intercept and the distance-rate-time equation would look like:
Which looks very much like:
Notice that r in the distance equation is analogous to the m in the slope-intercept equation? No wonder that the slope is also known as the rate of change of the function.
The only difference between graphing a line such as and graphing something that looks like is that you need to take some special care with the scaling of the units on the axes. For example, if you were examining what happens in a trip that lasts 3 hours at a rate of 60 mph, the distance axis would have to go from 0 to 180 while the time axis would only go from 0 to 3.