![]() ![]() With that in mind, one of my goals was to provide my students with activities that truly fostered authentic work. With our current technology, students have access to step-by-step solutions at their fingertips to almost any problem we give them. This past school year, as many other teachers, I taught in a hybrid model with some students in person and some others at home. Recently I went through my old Desmos ( ) graphs and I’d like to share 5 of my favorites here.Working as a math educator for the past twelve years has taught me that an effective way to engage students is by giving them choice in how they demonstrate mastery of their learning. There are some additional interesting graphs at the bottom that are cool as well. You can click on the links after each header to edit and interact with the graphs in the browser. The goal of this graph is to take a function $f(x)$ and rotate it around the origin by an arbitrary radian amount in order to demonstrate the concept of rotation in the complex plane. I accomplished this by first looking to rotations in the complex plane, then translating the method to the real plane. The blue curve is the original function and the orange curves are the rotated versions.Ī complex number $z=x yi$ is plotted in the complex plane by putting the real part $x$ on the horizontal axis and the imaginary part $y$ on the vertical axis. Note that this means each “point” is really just a single number. This means a $z$ can be rotated about the origin $a$ radians by multiplying by $i^$) and $\theta$ is the angle of the same vector with the positive x axis. Unfortunately, Desmos does not support complex numbers. ![]() ![]() Luckily, the parameterization of the complex unit circle is exactly the same as the parameterization of the unit circle in the real plane (excluding the imaginary unit of course). In the Desmos graph I took advantage of this fact to rotate each point. To rotate an entire curve (as opposed to a single point), I made a parametric function of $t$ and applied the transformation to each point $(t, f(t))$ in the domain $a\leq t \leq b$. I like this one because of the simplicity. I also like it because I had no idea how to do calculus when I created it about a year and a half ago. However, I did understand that the derivative magically yields the slope of the function at any given point. So I decided to create a little demonstration, and this is probably one of the first things that got me seriously interested in math. The black curve is the function (you can edit it from the link above), the red line is the tangent, and the blue line is the normal. This graph shows the functions that yield the $x$ and $y$ coordinates of a point as it moves around an ellipse. This is analogous to the $\sin$ and $\cos$ functions for a circle. In fact, if you set $a=c_0$ and $b=c_0$ in the graph above for some $c_0\neq 0$, the resulting functions will be the sine and cosine. The blue curve is an analog of cosine, and the red is an analog of sine. I don’t know of any applications for this, but it is pretty interesting to see how the shape of these functions change depending on the characteristics of the ellipse. For example, from the image above, you can tell the major axis is in the $x$ direction because the cosine analog has a greater range than the sine analog. The top and bottom of the blue curve are pinched together because the direction of the point moving along the circle is rapidly changing since the ellipse is wider than it is tall. ![]()
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