# Tag Info

14

Just supplementing Benjamin Dickman's nice answer, here is $x \mapsto x^2 - x$ in $\mathbb{Z}_{18}$ in the same style: For example, the pentagon wheel reflects the fact that $$(5+3k)^2-(5+3k) = 9k^2 + 27k + 20 = 9k(k+3) + 20 = 2\bmod 18 \;.$$

12

One of the approaches taken in some areas of mathematics (e.g., in arithmetic dynamics and considerations of preperiodic points, etc) is to create these graphs by drawing discrete points and then using arrows to show which values map to which other ones. Figuring out a "canonical" way to draw these pictures might be a bit tough (this is related in some ...

9

Knowing basic graphs is a simple way to geometrically understand a wealth of data about functions. If a student understands what a point on a graph means as it relates to a function then there is so much that can be remembered by a simple picture. For example, $\sin \theta = \frac{1}{2}$. For the careless calculator user, they'll answer $\theta = \pi/6$ or ...

9

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. ...

8

By replacing hand-graphing with computer graphing, we place a black box between equations and their graphs in the experience of our students. Black boxes are bad for understanding. In fact, this is the entire point of a black box - to create a level of abstraction that frees a user from the gritty details of some particular task. Obviously there are times ...

8

This is pretty straightforward in Mathematica: f[x_] := x^2; xrangeleft = -2; xrangeright = 2; yrange = {0, 4}; xrange = {xrangeleft,xrangeright}; frames = Table[ListPlot[Table[{x, f[x]}, {x, xrangeleft, xrangeright, 1/2^i}], PlotRange -> {xrange, yrange}], {i, 0, 10, 1}] Export["FunctionPlot.gif", frames] (You might need to use Directory[] to ...

7

My favorite example of a graphical transformation is waves. Consider, $$y = A \sin( x-vt)$$ Here $A$ gives the amplitude of the wave and it tells us from a graphical perspective how much the unit-sine wave is vertically stretched. If we fix $t$ then the term $-vt$ is just some phase-shift and we can see the graph is just a horizontal shift of the sine wave....

7

Unit conversions often provide a natural application of these transformations. Since temperature unit conversions often include both scaling and shifting, they are particularly useful. For example: Suppose you have a function $H(c)$ which gives the rate of heat stroke death in a region, per thousand occupants, as a function of the temperature $c$, in ...

6

Asymptote - if $f(x)-g(x)\to0$ as $x\to a$, then $f(x)$ is asymptotic to $g(x)$ as $x\to a$. Similarly, $g(x)$ is asymptotic to $f(x)$ as $x\to a$. (IMO) Personally, I do not see the problem with intercepting any amount of times. If you wish to modify the definition of an asymptote to fit this, I would probably use limit superior and limit inferior. ...

6

I once worked in a factory that used "six sigma" to improve its production processes and yields. "Six sigma" involves a lot of graphs. In half of the factory, the manufacturing engineers used computers to prepare blank charts, and the line workers drew the graphs in the charts themselves. In the other half of the factory, the manufacturing engineers used ...

6

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

6

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. ...

5

I don't think it's all that bad to put your graph on ordinary everyday axes as long as the students know that the order is more or less irrelevant. If you are happy to break out of the page, I recommend drawing your graph on a piece of paper and rolling it up to make a cylinder. Then at least one of the axes represents the cyclical nature of the field. If ...

5

Edited: I would use a rectangular display that looks, at first, like a standard "Quadrant I" graph, but that can be grabbed and dragged left/right/up/down to move the viewing frame. So, for example, if one is working over $\mathbb{Z}_7$ the horizontal and vertical scales will initially be labeled "0 1 2 3 4 5 6" in both direction, but the view can be ...

5

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 ...

4

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 ...

4

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.

4

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 for $y=x(x−1)(x+1)$:  &...

3

Here is the code in Matlab for the sine function: for k = 0:6 n=2^k % stage number, k=0: integer values, k=1: stage 2, ... x=0:1/n:3*pi; y = sin(x); % sine function pause(1); % pause duration plot(x,y,'b.') axis ([0 3*pi -1.5 1.5]) grid on end

3

There are dangers in using such graphical representations. The reals are an ordered field, whereas orders are not compatible with field operations for finite fields and other fields. It is easy to think things like "this function is increasing": such thoughts are helpful if you are going to transfer something of the function to the reals, and can mislead ...

3

Students are expected to graph a quadratic equation, a parabola, and indicate on the graph the five points of interest - vertex 2 intercepts y-intercept symmetric point If you graph this on a computer or smart phone, you are given the answers, same as plugging in the equation and asking a calculator to spit out the two solutions. I can't cite a study, but ...

3

You might try getting the students to understand Braess's Paradox, which can be phrased as a paradox about traffic flow, modeled by a weighted graph. The paradox is that the addition of a "short cut" road leads, under individual rational behavior by each driver, to everyone taking longer to reach their destination.           The ...

3

Perhaps an approach that mirrors the standard graphing might be useful? One way that I appreciate is seen here: This is used by N. J. Wildberger and others. I just snagged this off google images to demonstrate. I think this particular image is of $F_{13}$ with two "lines" plotted and their intersection marked at (5,2) -- but don't quote me on that. I can ...

3

I don't think that a horizontal line being asymptotic to itself is a "trick question we just never really run into" (as you suggest in the comments). Probability theory gives a natural context in which this sort of thing occurs (albeit in a way which is more than just a horizontal line). A standard result is that a continuous function $F$ defined on $(-\... 3 Mathematically a horizontal line is asymptotic to itself. So for example the horizontal asymptote of the constant function$f(x)=3$is the line$y=3$. This issue of whether the definition of a horizontal asymptote should place some sort of restriction on the graph of$y=f(x)$crossing a horizontal asymptote seems to come up frequently. As a mathematician ... 3 Desmos may be less powerful than geogebra, but is worth mentioning in this regard. Graphs on desmos are very easily animated (with sliders), and look great with little modification. You can find much better examples, but here's a very simple one I made for thinking about the tangent line to a curve. 2 Calcplot3d might be a good choice: http://web.monroecc.edu/calcNSF/ 2 If we open up graphs to include infographics there's little you can't communicate, and when you say that you want to communicate insight that's what comes to mind. I don't have much insight to offer on creating complex infographics. Read some Edward Tufte he's a leading expert in compact informative graphics. The Grammar of Graphics by Leland Wilkinson is ... 2 If I may suggest a tiny variation on Aeryk's nice code, to slow it down, and join the dots: f[x_] := x^3; xrangeleft = -2; xrangeright = 2; yrange = {-8, 8}; xrange = {xrangeleft, xrangeright}; frames = Table[ ListPlot[Table[{x, f[x]}, {x, xrangeleft, xrangeright, 1/2^i}] , Joined -> True ... 2 Imagine the simple equation of a straight line:$ y=2x$. You want to move this up one point (in the$Y$-direction), and you get:$y=2x+1$. If you want to move this up one point in the$X$-direction, then you first need to write this with$x$at one side:$x=y/2$. So you get :$x=y/2+1$. Writing this back to the$y=f(x)$notation becomes:$y=2(x-1)\$. So, ...

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