Modeling is one of the big themes of the AP Precalculus course. A subskill of this larger category is being able to write an equation that passes through particular points. Students focus on this particularly with linear and exponential functions, since these functions have the unique property that just two ordered pairs determine the equation.Â

Now that students are at the end of the course and have familiarity with *all *the function types, why not up the ante? In the **Chili Pepper Function Challenge** students are given two points on the coordinate plane, and each challenge asks them for some kind of equation that passes through the points, where the function type is specified by the challenge. Students will also be asked to write alternate but equivalent forms of equations theyâ€™ve written in previous challenges.

Why the chili peppers? Some of these challenges have some heat! Each challenge is rated based on its spice level, from one to four chili peppers. Students must collect fifteen chili peppers in total, but can choose which challenges they would like to complete. This is a great way to differentiate and at the same time foster studentsâ€™ autonomy.Â

## Tips

We strongly encourage students to work in pairs or small groups on these challenges, as they are difficult! Students should **not **be using the regression features on their calculator. These challenges should be completed without a calculator, though we do recommend having students check their equations using Desmos to verify that they do in fact pass through the given two points.

Note that several of the challenges will have several correct answers. The solutions we post are thus simply sample student answers. Desmos is the easiest way to check if a certain equation meets the requirements. You could even have students submit a screenshot with their equations and graphs, demonstrating that they pass through points A and B.

I believe b in Challenge #3 is incorrect, the function is exponential decay so b should be less than 1.

I'm using this with my AP precal kids this year, I think it'll be great!!