{"id":4115,"date":"2010-02-02T08:35:00","date_gmt":"2010-02-02T00:35:00","guid":{"rendered":"http:\/\/www.intmath.com\/blog\/?p=4115"},"modified":"2019-09-20T17:06:58","modified_gmt":"2019-09-20T09:06:58","slug":"graphs-using-free-math-software","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/learn-math\/graphs-using-free-math-software-4115","title":{"rendered":"Graphs using free math software"},"content":{"rendered":"<p>[<b>Updated Dec 2016.<\/b>]<\/p>\n<p>Being able to sketch graphs on paper is an important skill, but I'm wondering about the future of this. There are now many free (or cheap) graphing tools available and I am surprised so few students actually use them (or are allowed to use them).<\/p>\n<p>Let's have a look at some of the available math graphing tools. <\/p>\n<h4>Graphics Calculators<\/h4>\n<p>Here are the Texas Instruments TI-83 (left) and Hewlett-Packard HP 40gs (right) calculators. <\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/TI-83.jpg\" alt=\"TI-83 calculator\" width=\"127\" height=\"257\" \/> <img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/HP.jpg\" alt=\"HP calculator\" width=\"127\" height=\"257\" \/><\/p>\n<p>Graphics calculators are handy, but the screen size is small and there is usually no scale on the axes. Cost is often prohibitive, at around US$100. <\/p>\n<h4>Online Graphers<\/h4>\n<p>Here's a few places where you can graph your curve - for free! <\/p>\n<p>My favorite in this space is:<\/p>\n<ul>\n<li><a href=\"https:\/\/www.desmos.com\/calculator\">Desmos<\/a><\/li>\n<\/ul>\n<p>Don't miss <a href=\"http:\/\/dailydesmos.com\/\">The Daily Desmos<\/a> for some cool graphs.<\/p>\n<p>The next example uses Scalable Vector Graphics, right here on IntMath.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.intmath.com\/functions-and-graphs\/graphs-using-svg.php\">Plot your own SVG Math Graphs<\/a> <\/li>\n<\/ul>\n<p>For another grapher based on a well-known SVG grapher, see: <a href=\"https:\/\/www.intmath.com\/functions-and-graphs\/graphs-using-jsxgraph.php\">JSXGraph function plotter<\/a>.<\/p>\n<p>The following are also free offerings (some suggested by readers). I've presented them roughly in order of preference.<\/p>\n<ul>\n<li><a href=\"http:\/\/www.wolframalpha.com\/\">Wolfram|Alpha<\/a> (produces a graph image, which can be converted to CDF interactive if you have the pricey Mathematica)<\/li>\n<li><a href=\"http:\/\/graphsketch.com\/\">GraphSketch.com<\/a> (image-based)<\/li>\n<li><a href=\"http:\/\/goodcalculators.com\/\">Good Calculators<\/a> (canvas- and javascript-based)<\/li>\n<li><a href=\"http:\/\/www.graphfree.com\/\">GraphFree<\/a> (image-based)<\/li>\n<\/ul>\n<h4>Downloadable software apps<\/h4>\n<p>Here are some of my favorites. In each case you need to download and install the software. <\/p>\n<ul>\n<li><a href=\"https:\/\/www.geogebra.org\/\">GeoGebra<\/a> (Free. I wrote a <a href=\"https:\/\/www.intmath.com\/blog\/mathematics\/geogebra-math-software-a-review-2-10096\">review of GeoGebra<\/a> which shows how to get started with it.)<\/li>\n<li><a href=\"http:\/\/download.cnet.com\/Graph\/3000-2053_4-10063417.html?tag=mncol\">Graph 4.4.2<\/a> (Free. Small download, from cnet.com)<\/li>\n<\/ul>\n<p><b>Requires payment:<\/b><\/p>\n<ul>\n<li><a href=\"http:\/\/www.mackichan.com\/index.html?products\/snb.html\">Scientific Notebook<\/a> (US$90, but does much more than a graphics calculator. It's a full computer algebra system, but much cheaper than MatLab or Mathematica, etc.) <\/li>\n<\/ul>\n<h2>Some problems with using graphics software <\/h2>\n<p>If you don't have a good idea of what a function should look like before your use a graphics package, you can have all sorts of problems.<\/p>\n<p>Here's an example of a function which catches out the unaware.<\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/30overx-2.gif\" alt=\"30\/(x - 2)\" width=\"86\" height=\"38\" \/> <\/p>\n<p>Let's graph it on Geogebra. <\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/geogebra-1.gif\" alt=\"Geogebra 1\" width=\"195\" height=\"179\" \/><\/p>\n<p>Hmmm - it appears to be empty. Is there an error? What's going on?<\/p>\n<p>We <strong>zoom out<\/strong> a few times and start to see 2 curves. Once again, is there a mistake? Why 2 curves? <\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/geogebra-3.gif\" alt=\"zoomed out\" width=\"327\" height=\"322\" \/> <\/p>\n<p>I zoomed out because I had a good idea in my head what the graph should look like. Since I couldn't see either of the 2 arms of this curve, and because of the 30 in the denominator, I knew I had to zoom out. <\/p>\n<p>If I left my graph at that, I would still not have a good idea of what the function looks like. I have not chosen a view that shows the crucial features of this graph. <\/p>\n<p>The <strong>default view<\/strong> in Geogebra was too close to the origin (0, 0) to see any of the curve. <\/p>\n<p>Let's have another go, this time using Scientific Notebook. This is what I get when I try to graph the curve. <\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/SNB-1.gif\" alt=\"SNB\" width=\"317\" height=\"229\" \/><\/p>\n<p>I get a vertical line passing through 2 on the <em>x<\/em>-axis. But notice the scale on the <em>y<\/em>-axis. The number &quot;4e+09&quot; actually means 4 &times; 10<sup>9<\/sup> or 4 billion. So Scientific Notebook recognizes that there are very large values of <em>y<\/em> involved in this function, and has shown us the limits of its internal coding.<\/p>\n<p>This time if we <strong>zoom in<\/strong>,  we can start to see the graph appearing as before. But I need to <strong>know<\/strong> to zoom in, otherwise I would miss it altogether. <\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/SNB-2.gif\" alt=\"SNB-2\" width=\"329\" height=\"236\" \/><\/p>\n<p>What was that vertical line that appeared before? Was it a mistake? Why did it disappear when we zoomed in?<\/p>\n<p>Next, we use my <a href=\"https:\/\/www.intmath.com\/functions-and-graphs\/graphs-using-svg.php\">SVG Grapher<\/a>. It's similar to Geogebra in that its default view misses the curves, but is also similar to Scientific Notebook in that it shows a vertical line through 2 on the <em>x<\/em>-axis.<\/p>\n<p>When we zoom out a bit, this is what we get. <\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/SVG.gif\" alt=\"SVG\" width=\"350\" height=\"287\" \/> <\/p>\n<p>We are still not showing all the vital features of the graph, and that vertical line is still there. What's that about?<\/p>\n<p>Next, I tried the  Flash Math Grapher. Once again I needed to do some zooming. The <em>x<\/em>- and <em>y<\/em>- scales are strange (multiples of 13.01??, and is that the <em>x<\/em>-axis or the line <em>y<\/em> = -2.98?) , but at least I can get a pretty good view. <\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/flash-math-grapher.gif\" alt=\"Flash Math Grapher\" width=\"428\" height=\"352\" \/><\/p>\n<p>Next, let's call in the big guns. <a href=\"http:\/\/www.wolframalpha.com\/\">Wolfram|Alpha<\/a> gives us the following 2 graphs when we put our function in their search box (and they give us a lot more information about the function).<\/p>\n<p>This time we get intelligent graphs that have appropriate <em>x<\/em>- and <em>y<\/em>- scales and actually show the curves. Note the first one does not have a vertical line through 2, but the second one does. Why? <\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/wolfram.gif\" alt=\"Wolfram\" width=\"234\" height=\"300\" \/><\/p>\n<h2>That vertical line through <em>x<\/em> = 2<\/h2>\n<p>When the software draws the graph, it chooses regularly-spaced <em>x<\/em>-values and substitutes them into the function, plots the resulting dots and joins them.<\/p>\n<p>In the above examples that have a vertical line, it means the software has chosen a value just slightly less than 2 (which will give a very small, negative value for <em>y<\/em>) and another one just slightly more than 2 (which gives a very large positive value for <em>y<\/em>).<\/p>\n<p>Here's an exaggerated version of what it's doing (using only a very few data points joined by straight lines).<\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/geogebra-5.gif\" alt=\"exaggerated\" width=\"456\" height=\"286\" \/><\/p>\n<p>If the <em>x<\/em>-value chosen is exactly 2, the software will normally just skip that value (since it will return a &quot;division by 0&quot; error). The (almost) vertical line should not be included in the graph, since we cannot have <em>x<\/em> = 2 (this function is undefined for <em>x<\/em> = 2 since the bottom of the fraction would be zero). <\/p>\n<p>This gap in the graph is called a <strong>discontinuity<\/strong>. It should be a gap - not joined by a line as above. <\/p>\n<p>Some software handles this situation gracefully (like the Wolfram|Alpha example) while in others, you need to either understand why the vertical line is there, or in some cases, you can elect to include discontinuities or not. <\/p>\n<h2>Best answer<\/h2>\n<p>Here's probably the best way to display the graph of this function.<\/p>\n<p>Our graph has 2 <strong>asymptotes<\/strong>. When a curve gets closer and closer to a line but does not touch it, that line is called an <strong>asymptote<\/strong><\/p>\n<p>The first asymptote is the <em>x<\/em>-axis and the other is the vertical line, <em>x<\/em> = 2, which I drew using a dashed line of a different color (since it is not part of the graph). <\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/geogebra-4.gif\" alt=\"final answer\" width=\"479\" height=\"466\" \/> <\/p>\n<p>I have shown the 2 asymptotes clearly and I have also labeled the <em>x<\/em>- and <em>y<\/em>-axes.<\/p>\n<p>Here's the function again for convenience.<\/p>\n<p><img loading=\"lazy\" src=\"\/blog\/wp-content\/images\/2010\/01\/30overx-2.gif\" alt=\"30\/(x - 2)\" width=\"86\" height=\"38\" \/><\/p>\n<p>I knew the graph was going to involve asymptotes since the <em>x<\/em>-variable is in the denominator (and we can't have 0 in the denominator) and also considering when <em>x<\/em> gets really big, the value of the function will be really small.  <\/p>\n<p>If I didn't know this (from graphing many of them on paper), I would have made a mess of graphing my function on a computer. <\/p>\n<h2>Graphing software and the future<\/h2>\n<p>Will graphing software change what we do in classrooms? Should it? Is it really necessary to sketch graphs on paper still? <\/p>\n<p>If you can draw a quick sketch of a function on paper, it certainly helps your understanding for many types of math problems. The conclusion from above is that it is certainly worthwhile to have a good sense of what a graph should look like before graphing it using software, so we can manipulate the settings to show the graph properly.<\/p>\n<p class=\"alt\">See the <a href=\"https:\/\/www.intmath.com\/blog\/learn-math\/graphs-using-free-math-software-4115#comments\" id=\"comms\">23 Comments<\/a> below.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Do we still need to graph on paper, or should we use free math graphers instead?<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mo_disable_npp":""},"categories":[102],"tags":[134,127],"_links":{"self":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/4115"}],"collection":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/comments?post=4115"}],"version-history":[{"count":5,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/4115\/revisions"}],"predecessor-version":[{"id":12169,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/4115\/revisions\/12169"}],"wp:attachment":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/media?parent=4115"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/categories?post=4115"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/tags?post=4115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}