What are some tips for framing a graph in the most readable, relevant, and aesthetic way, for secondary education mathematics?

Question

When I say "framing," I mean things like choosing zoom, x-axis/y-axis step, horizontal/vertical shift from the origin, choosing how/when to number steps, labeling axes, as well as, purely aesthetic details like using a grid (or not), labeling the axes "x"/"y", colouring/texturing the graphed functions, thickness of the lines, resolution, pixel dimensions, and so on.

Secondary education mathematics like algebra, geometry, trigonometry, precalculus, calculus. So lots of polynomials, trigonometric functions, logarithmic and exponential functions, rational functions, conics, functions in polar coordinates, parametric equations/curves, etc.

An example I can already think of is: if you're graphing a conic, it's a good idea to frame the graph in such a way that the student can clearly read the center/foci/vertices/etc because identifying them will likely be a necessary or helpful task in understanding the material.

Answer 1

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. In this case, you can ask they highlight those 2 points on the line.)

Parabola - Typical to ask for "five points" - Vertex, 2 X-intercepts, Y-intercept, and symmetric point. [Note, this is the generic approach, of course if the parabola doesn't intersect X-axis, those 2 points don't exist.] Focus/directrix for the higher level.

Cubic - The X-intercepts, and Y-intercept if appropriate. Let me offer a graph of a potential problematic situation.

$y=\left(x-5\right)\left(x+1\right)\left(x+8\right)$

I say "problematic" for one common issue I see. A teacher might give a worksheet/exam where the blank graph has a scale already printed. That scale can be 10X10 or 20X20 and it would cover the 3 X-intercepts, but not let the student draw the local extrema. The request sometimes feels ambiguous to the student. A 'sketch' might mean general shape of the curve and identify X-intercepts, along with Y (with a note on the graph, indicating "-40"). For advanced students, usually pre-calc, the exact extrema values are important as well. And note that to show them in a pleasant graph, the scales are different. I'd prefer graphs with unlabeled axis for the more advanced students' use.

For the other conics, circle/ellipse, hyperbola, a similar approach. These have their own points of interest, foci, vertex, directrix, asymptotes, and it's important to have problems be clear, i.e. if they are given the graph, those points should be labeled or identified, if they are given the equation, their graph should should these points and the drawing itself be neat.

Trig graphs - There are a number of directions problem come from. "Graph this equation", "Look at this graph and write the appropriate equation", "Here is a word problem. Graph the sinusoidal function, and give the equation."

In each case, I'd suggest (a) Identify the minimum and (b) maximum values, as well as the (c) sinusoidal axis / midline. Next, Identify the (d) period. From there, the appropriate starting point (the horizontal shift) is determined and the graph drawn. For some problems, one period is enough, others, multiple periods. I'd note that if min/max if a high enough number, the graph need not always show the X-axis, only the scale for x values. e.g. a value ranges from 100 to 110 over time. I've seen frustrated students produce a scale that has a blank area from 0-100, and a tiny graph above that. Better just to show them that not all graphs require starting at zero at either axis.

Answer 2

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 by and others only zoomed way out - or if you have an aspect ratio very different from 1. Consider the family of curves $x^4-nx^3$ for various positive $n$ and you will see what I mean, I hope.

I know that doesn't exactly answer the question, but I think it answers a possible question behind it about how to remove distraction from instruction surrounding graphing. And this is one possible answer to the "why are we doing this" surrounding careful plotting - it won't satisfy pure utilitarians, but it might be a nice way to say "the calculator did it" won't always work.

To bring it back to your actual question, I might recommend asking students to vote on different zooms, or to have a real-time zoom/frame choice if your computation method supports that. This may align closer to your pedagogical goals than just picking "the perfect one", at least after one's initial examples which should be simple enough not to have to worry about the choice.

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On a very practical note unrelated to (and possibly undermining?) my main comment, if you can get your plotter to plot $\sin(x)$ and friends with horizontal axis tick marks in (rational) multiples of $\pi$, that is a big win!