A fellow blogger, one who welcomed me warmly when I started, posted the following today: “Should students learn how to graph functions by hand?” His question is most intriguing. My comments to his post follow. If you have a special affinity for inverse functions, check out the link further below to a GeoGebra applet that is extremely powerful for use with your students.

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Hi David.

I am torn on this question as well, if access to technology is not an issue. If it is an issue, then I believe paper and pencil is the best choice for students outside of the classroom. If access is not an issue, I believe the important aspects of sketching any function are best introduced using some form of technology. I prefer GeoGebra as you use above, although there are may fine alternatives.

Afterwards, once students clearly understand the key concepts such as slope, y-intercept, x-intercepts, axis of symmetry, asymptotes, holes, extrema, amplitude, phase, periodicity, etc., depending upon the level of the mathematics, they should be able to apply their understanding manually (paper and pencil / pen, whiteboard and marker, etc) to simple functions, as that is still the most prevalent means of communicating in small groups. More complex functions are typically modeled using software due to their difficulty.

While writing this comment, my son asked me a question about finding the domain of a function such as y = sqrt(x^2+5x+6). While I explained it to him algebraically initially, I elected to show him graphically using GeoGebra to cement his understanding. I could have sketched it manually for him, but even this simple function would have taken too long.

Also, a couple of months ago, a new pre-calculus teacher asked me if she should spend time having students learn how to graph various functions manually. My advice was for her not to have students spend too much time using manual graphing methods to discover the key concepts. Rather for her to introduce, or review, them using software, then have students work with both manual methods and software. Unfortunately, access is a challenge still for most students, even if money is not an issue. The need to install the software and develop adequate proficiency with it still seems to challenge high school students, so students end up manually graphing on most problems anyways. Even calculus textbooks that incorporate uses of CAS or graphing calculators on occasion throughout still emphasize manual methods for curve sketching using sign charts, etc.

For a specific example of the power of technology to aid in teaching graphing, I sent the following link, and comments, to the pre-calculus teacher I mentioned earlier stating it is an excellent tool for exploring how the inverse of a function relates to a function, and vice versa

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You must try this out. It is an eye opener for students learning (and relearning) the inverse of a function. Be patient when going to this url. It takes a few moments to load in your browser.

The software running this tool is GeoGebra, which I have shown students several times. This specific “app” allows you to input any function and simultaneously observe its graph and the graph of its inverse. You can also observe how the inverse function becomes a reflection of the original function over the line y = x, and restrict the range of the output. Lastly, it allows you to note how two points, one on each graph, are inverses of one another, where one is point A at (x,y) and the other is point B at (y,x). http://www.geogebratube.org/student/m3211

Also, note that you can change the function to be any f(x) you wish. Start out with a constant such as f(x)=5, then try a negative slope line such as f(x) = -x, then quadratics, cubics, exponentials: f(x) = e^x, rationals: f(x) = 1/x, trigonometrics: f(x) = sinx, etc.

You will radically improve your students’ understanding of inverse functions with this tool!

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Dave

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