# Rational Power Functions

One of the topics our PreCalculus syllabus (Math 111) covers is “Rational Power Functions.” Since functions like $f(x)=x^n$ are called power functions, the rational power functions would be those of the form $f(x)= x^{p/q}$ (where $p/q \in \mathbb{Q}$ ). Unfortunately, this topic isn’t covered in our textbook (Zill’s “Essentials of Precalculus with Calculus Previews“).

Tom Kunkle, on his Math111 Homepage, provides his students a nice summary of how these functions behave: See ratpowfunc.pdf. When thinking about this week’s Lab Assignment, my goal was to give students practice on drawing graphs of functions like

$y = -(x-5)^{4/3}+1$
Lab Description
I created nine cards, which I’m calling “Function Cards.” Each card is a half-page (printed on heavy-duty card stock). On one side, the card has a Graph of some translated rational power function. On the other side, the card has an Equation of a translated rational power function. However, the equation and the graph are not the same function!

Here are the directions I provided to students:

1. When you get your Function Card, flip it to the Equation side. Make a note of the card number. Write down the equation.
2. As a group, work together to sketch a graph of the equation. You may use the Rational Power Function handout (if you like). You may NOT use a calculator. Work together! Your sketch should clearly display & label all intercepts, any cusp points, any vertical asymptotes, any locations where the tangent line is vertical, and the parent function.
3. Once your group has agreed upon what you think the graph looks like, draw your sketch on the Answer Sheet, along with the information listed in Step 2. Then go around to the other groups. Find the Function Card with the graph that matches your sketch. Ask nicely, then take it from that group.

The students will have the entire class period to create graphs of each of the nine Functions. Since they will track down the graph most closely matching theirs, they will have a chance to check their answers. Even once they see the answer (i.e., the graph), they are still asked to do some thinking since they have to provide information about the graph’s important features.

Hopefully this process will work. The only thing that concerns me is how long it will take them. I am really terrible at gauging how long it takes students to think about things. My best guess was that it would take about 4 minutes to sketch a quick graph, and then another 3 minutes to write down its important features. So, (7 minutes)x(9 Function Cards) should be a little over an hour.

I picked the nine functions from Tom’s “Homework Handout” on this Section, available here. If the weather complies, maybe we can do this activity outside. The only thing better than math class is math class in the sunshine.

# First Day Activity

I really enjoyed today’s “First Day” activity in Precalculus. I found the idea on Becky Lyon’s blog; you can also find her on Twitter: @rhlyon.

I had the students find someone to work with and told them one member of the pair would be the Explainer and the other would be the Grapher. The Explainer was supposed to sit facing the projector screen, while the Grapher was supposed to sit facing the door (i.e., away from the projector where they could not see it at all).

The idea of the activity is this:

1. Display a picture or graph on the projector screen for about one minute.
2. The Explainer has to describe the graph only using words — no hand gestures allowed!
3. The Grapher tries to re-create the picture or graph from the description.

The pictures I used started out easy (a giant smiley face) and got progressively more difficult. To give you an idea, I uploaded the exact graphs I used to my public Dropbox space: It’s http://dl.dropbox.com/u/59433434/111-Day1.pdf. After we were done, we went through the graphs together and talked about what descriptions had been given and what people could have said to make it easier.

This gave us a great opportunity to review vocabulary like “degree” and “vertex” and “parabola” and “quadratic” and “intercept” and “slope” and “local maximum [versus global maximum]”. It also gave me feedback as to what the “groupthink” occurred and at what level my students are starting. (For instance, some of them volunteered the idea that an even-degree root of a polynomial behaves differently on a graph than an odd-degree root!)

They seemed to enjoy the activity. It helped cement for them that I will expect them to do things in class, not just be. And, happily, it gave them the opportunity to practice my Friendship Policy.

My three favorite comments from today include:

• This will be the most FUN class!” –a student said to her friend, at the end of class
• An e-mail I received after class said, “What a great first day of class! Super exciting and thank you for your approach!
• A fantastic Tweet (admittedly from a calculus, not precalculus, student):