When I look at most types of functions in two-dimensions, I can more-or-less visualize what is happening. This is a byproduct of working with these functions for over seven years. At one point, you start to get a feel for how a parabola will look, or what placing a factor in the denominator of an ellipse will stretch or squeeze the shape. However, when you first start learning these functions, they aren’t immediately obvious.
Unfortunately, many students have problems with this (which is natural), but the bad thing is that many teachers don’t address this issue. I see so much potential for students to better understand how functions work and how changing the parameters affect the shape of the graph. With the tools we have today, I think every teacher should be using them in order to help solidify their students’ understanding, particularly at the beginning.
The tool I’m going to talk about here is Desmos (Link), but of course if you know another tool that works, then go for it. However, there are two real key features of Desmos that I would say put it above the rest in terms of graphing. For one, it’s buttery smooth. It doesn’t take five minutes for a function to be plotted. The function is updated as you write, which is fantastic. The real feature that I love though is the use of sliders. This is precisely what can give students a feel for a function.
Let’s suppose we have a standard form quadratic function:
For most graphing software, you need to specify what all the constants are. However, you can simply write them as variables in Desmos, and it will create sliders where you can dynamically change the constants and see the consequences on the graph. Additionally, you can specify the range of values for each slider, and by what step size it will increase/decrease every time. Once you have those settings as you wish, you can either manually move the slider or have it “play” through all the options.
When I first used the power of this specific feature, I was studying polar curves in calculus III. I used the sliders because I was interested in seeing how polar roses or cardioids would change as the constant parameters were altered. Desmos became a great help for gaining that sort of intuition which helps even when I don’t have an instance of Desmos running.
The point is not to create a dependence on graphing software for the students. Instead, it’s about giving them a foundation in which to base their intuition. Intuition is not a magical ability to understand mathematics. It comes from visualizing and deeply understanding the underlying principles that are at work. In that sense, I want students to be capable of getting that visual aspect of functions and graphs that is so often missing. I want them to understand what changing $a$ in the above equation does to the function.
I am now trying to use this great tool with all my students I tutor. I think it is quite obvious that using graphing software to dynamically show how a function’s graph changes when parameters are altered is much more interesting and easier to remember than just being told that increasing that $a$ symbol on the page will make the parabola narrower.