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Most likely TikZ. Rather time-consuming to learn.

The slope field during the exam was not wrong. The slope field with the swapped x and y axis was a recreation that I had requested someone on ConquerHSC to make for me so that I could write up a set of solutions, since I only had a poor quality photo version of it. They recreated each dash on...

I wrote the set of solutions that you are referring to here. As for Q12a), I think NESA should accept both solutions. A slope field is not sufficient to define the equation of a line passing through a point as there is not enough detail on the slope field to accurately plot it. I can see why...