The students are given four points in a rectangular coordinate system. They are to fit the points to an exponential equation of the form y = a* b^(x-h)+ k. Two of the equations are exponential growths ( b > 1), and two are exponential decays (b < 1). To solve this project, the student must use some deductive logic to determine the steps and the order of these steps to derive the equations that fit the given data (four coordinates). This involves the solution of four equations with four unknowns (although it is a bit more difficult than just a system of equations) and the knowledge of exponential and logarithmic equations.
The typical high school algebra 2 and precalculus textbook describes how to write the equation of exponential and logarithmic equations, but most use the simplified equation form y = a*b^x, and one of the points is always at x = 0 (which is the value of a).
This project is a challenge for all students and many will be unable to complete the project without guidance. For this reason, two forms of each of the four projects are provided. You, the teacher, must decide which form is best for your class(es). The forms should not be mixed in the same class. Most classes will need the guided form of the project. An advanced class should be able handle the unguided form.
Example of an Exponential and Logarithmic Project
Given: a 1/2.
(x1, y1) = (-2, -1/2)
(x2, y2) = ( 0, -2 )
(x3, y3) = ( 0. -7/3)
(x4, y4) = ( 1, -2 5/9)