Two problems to explore with graphing software

In preparation for a session for teachers designed to improve their confidence in using graphing software, I came up with two problems as motivation. Try them yourself and then decide if they might be fun to use with your students. Personally, I’d run the first one in Core 3 and the second one in Core 1. Let me know your ideas for varying them or other problems to try!

Problem 1: Plot the four points (1,3) (3,-1) (4,1) and (6,3). Interpolate a value for when x=2.

I think there is a lot of scope for using the language of functions within this investigation. What if the data must be modelled using a periodic function? Or one that is odd/even? Or with certain vertical asymptotes?

This is the perfect candidate for using constant controllers/sliders in your favourite program (Autograph, Desmos, Geogebra…)

Problem 2: The challenge here is using equations of straight lines with integer coefficients. Construct three such straight lines that bound a triangular region containing the point (3,2) in its interior. How small can you make the area of this triangle?

This is a good candidate for learning some basic tools in Geogebra: finding points of intersection of lines and creating a polygon. Requiring the coefficients to be integers I think forces more mathematical understanding than would otherwise be necessary to make the area ever smaller.