The problem underlying the discussion in the question can be summarized as that it is necessary to choose a branch cut to define a complex logarithm (or arctangent).
It is a mistake and pedagogically a bad practice to allow negative values of $r$.
It is a mistake because pairs $(r, \theta)$ with $r$ possibly negative have no right to be called coordinates. ...
I think your last paragraph was most of a good answer. i.e. a graph should contain the points of interest.
Linear Equation - X and Y intercepts, and depending on the lesson topic, highlight points that were part of the problem as stated. (e.g. the problem statement may have been "given the 2 points, solve for the equation of the line and produce a graph. ...
1 hotdog = 100 calories and 20 grams of protein.
1 hamburger = 150 calories and 15 grams of protein.
Any meal can be plotted in hamburger/hotdog space or calorie/protein space.
You might also be interested in my answer here: https://matheducators.stackexchange.com/a/5788/117
Assuming you are teaching these topics, I think this is a perfect opportunity to show students why they need to know how to actually solve these things - because there is no "one answer" for what zoom/min/max is best.
For some more complex functions there will be no one frame that will even show all significant features, as some may only be visible close ...
Edit (June 2019): I used this final project with minor tweaks again this year. You can find links to some of the output for Spring 2019 - both students' graphs and write-ups - here and here.
I used a Desmos Make-A-Graph prompt for an Algebra 2 class' final project last year. The results were quite good; so, I expect to incorporate at least one similar ...
The video to which you link is much more than a dynamic graph: It is an
entire lecture using many dynamic components, requiring considerable skill
with several technologies.
To return to a single dynamic graph,
in response to this MO question, Taking “Zooming in on a point of a graph” seriously,
I created this little animation
Somewhere in the comments somebody asked the same question and this link is the answer: https://talkingphysics.wordpress.com/2018/06/11/learning-how-to-animate-videos-using-manim-series-a-journey/
A somewhat easier way but far less powerful would be the program geogebra where you can export animations as gif.
Desmos may be less powerful than geogebra, but is worth mentioning in this regard. Graphs on desmos are very easily animated (with sliders), and look great with little modification. You can find much better examples, but here's a very simple one I made for thinking about the tangent line to a curve.
You mentioned 3Blue1Brown's videos. If you're interested in making similar animations, Grant (the man behind 3B1B) actually released the entire python library he wrote for creating them: manim on github. Obviously, this is very open ended and requires a working knowledge of python to begin with. The other examples of software (Desmos, Mathematica, etc) will ...
I tried two other ways along the graphical representation you are proposing.
A lookup table:
In the first line note some values for $x$: -2, -1, 0, 1, 2
In the second line note the corresponding values of $f(x)=x^2$: 4, 1, 0, 1, 4
In the third line note the corresponding values of $g(x)=f(x-1)=(x-1)^2$: 9, 4, 1, 0, 1
Fill them out together with the ...