Spreadsheet Math: Graph-i-pedia

Of all the activities I will share in this blog, none will likely undergo quite as much refinement as the one described in this entry. Although I did not make use of any spreadsheet program for this activity this year, I have written this entry to include the additions I have planned for next year.

Wikipedia is probably the single most used resource for information on any given topic. With the help of those knowledgeable in a given subject, information is collected and organized into the largest and most up-to-date encyclopedia in the history of the world. For the unit on functions and transformations of their graphs, I decided to have my students make their own “wikipedia” of graphs.

Part I

In groups of three or four, students were required to produce a series of pages dedicated to describing and displaying properties of various functions, which included

• linear function y = mx + b
• even polynomial y = x²
• odd polynomial y = x³
• square root y = √x
• absolute value y = |x|
• irrational function y = 1/x
• *any one trigonometric function
• *exponential or logarithmic

To start, I wanted my students to understand the effect that each constant factor added to an equation has on the graph of that function. Namely, if a function f(x) = ±Afunction(kx + h) + c, I wanted them to show the effect of the plus/minus, A, k, h, and c on the function respectively. For each function, they had to determine an equation that produced each of the following and provide its graph

• reflection -e.g. f(x) = -|x|
• dilation -e.g. f(x) =3x²
• horizontal shift -e.g. f(x) = (x - 5)³
• vertical shift -e.g. f(x) = √x + 4
• combination of all of the above -e.g. f(x) = -2/(x-3) + 4

DO THIS FIRST, BEFORE THE GRAPHS To help them further appreciate the effect of these changes on a given function, it is good to have them create a T-table and plot the points on a graph just as they would have learned to do in pre-algebra. However, because we will be working with so many different versions of a given equation, it is more efficient (and effective) to have them do this using a spreadsheet, where they can adjust the constants at will without having to change everything. This will demonstrate the effect on the corresponding outputs and then describe what they observe happens to each of the values.

A table of outputs for each transformation of a parent function.

After creating the table, students should then insert a graph (as shown in the Ohm’s Law and Making A Business activities) for each of the transformations. After completing this, they should then replicate their functions using the Desmos Graphing Calculator, screen shot it, and post it to their wikipage with a description of the graph (e.g. equation and name of transformations). Depending on how sophisticated one wants to be, they could also save their graph to their Desmos account and hyperlink it in their wiki.

To ensure each student complete this activity without simply copying from a group member or having “the smart kid” do all the work, I assigned each student one parent function and its corresponding transformations and only one (block) period to complete it. This time constraint forced the students to actively work on the project and help each other only conceptually as no one in any particular group would have the same graphs to produce nor the time to produce all of them.

Student samples of various transformations for an odd polynomial function. Note in the left sidebar that each function has its own page.

Part II

For each equation/graph containing the combination of transformations, students next had to determine: its inverse (and graph it), its continuity, end behavior, and critical points.

Regarding inverse functions, there are some interesting CONSEQUENCES that can be used to verify that the inverse was correctly obtained. The first of which is that the inverse should be a perfect reflection of the original function across the y = x line. Using Desmos, students can quickly check for this by adding an equation line and typing in “y = x” and see if the reflection is present.

Note that the red and blue lines are reflections of each other across the dashed y = x line signifying they are inverses of each other.

Another way they can verify their inverse is correct is to compose the functions together, the result of which would actually be they y = x line. While it would be worthwhile to have students check their result algebraically composing their functions in a new column of their spreadsheet, there are some inhibiting limitations that do not currently make this possible for all functions.

I will not be robbed of a good learning opportunity however -this is an excellent talking point for the emphasizing the importance of knowing how to check a result by hand. I would then immediately follow up with “don’t like it? then learn everything you need to in order to fix it and then actually do it because it isn’t going to get done by itself.” (I will post an update if I find a patch to make this happen).

Desmos will also automatically determine and display most critical points (max, min, intersections). However, I want to train my students to think about these points from the perspective of calculus -i.e. looking at the limit of f(x) as x approaches a particular value. In order to do this, we can again use our spreadsheet to confirm critical points as we approach them by changing the increments for the x value near a suspected critical point. But that will have to wait for next year.

Until then, here are some more student samples.