Great question.
The first thing I'd consider here is embedding the Desmos graphs as iframes. Many publishing platforms allow this kind of embed, and you wouldn't need to use an API at all. To do this, you can open your graph on
desmos.com, use the "share arrow" icon in the top right of the calculator, and choose "embed". This will show you some iframe code that you can paste into another web page. Note that some publishing platforms place restrictions on exactly how iframes can be embedded, so you might have to learn about those details for your platform.
The biggest thing to be aware of there is that that kind of embed will only show the graph, but not the equation list. There will be a link at the bottom right of the graph to edit the graph on
desmos.com that allows seeing the equations and editing them.
If you want students to be able to see/edit the equation list for a graph on your site , the API is the best way to go. You won't need to recreate the graphs. See this recent message for two possibilities (and some caveats) for getting data from a saved graph into the API:
One final comment is that loading _lots_ of separate calculators on one page, using either iframes or the API, is pretty resource intensive for students' devices. If you were thinking of putting, say, a dozen calculators on a single page, I'd probably instead go with putting screenshots on the page that link to saved states on
desmos.com, so that students can interact with the calculators one by one. That will be friendlier to low powered devices.
Best,
Jason