# AP® Precalculus – Change in Tandem

The College Board AP® Precalculus Course Framework contains many exciting ideas and novel approaches to familiar concepts. The centerpiece of any Precalculus course is the concept of a function. And AP® has a new, dynamic take on this familiar friend.

In the 1960s, when the idea of a function became a mainstay of school mathematics, it was often seen as a dry, theoretical concept. Back then, the curriculum focused on ƒ(x) function notation and the fact that a function is a set of ordered pairs.

### The New AP® Precalculus Course

Now, in the new AP® Precalculus course, a function is a dynamic mapping from input values to output values. These inputs and outputs change in tandem. A function is increasing if the inputs and outputs change in tandem in the same direction—that is, as x increases so does ƒ(x), or equivalently, as x decreases so does ƒ(x). Alternatively, a function is decreasing if the inputs and outputs change in opposite directions—as x increases, ƒ(x) decreases, or vice versa.

• One implication is that the course directs student attention to rate of change. In other words, what is the relative change in ƒ(x) compared to the change in x? This builds on the ideas of slope and rate of change, which students first met in middle school. We can answer questions such as, how fast is the car going? Is it moving forward or in reverse?

• We can take this a step further. Is the speed of the car increasing or decreasing? Is your foot on the gas pedal, or are you slamming on the brakes? It is natural to talk about and think about rates of change and even how these rates are themselves changing. And the AP® framework leverages this and includes such second-order rates of changes.

• The framework also features some new kinds of functions! For these, the input is not x, and the output is not y, or even ƒ(x). Polar functions have θ as the input and r as the output. Parameter functions have t as the input and both x and y as outputs. Closely related to these are vector-valued functions that have t as the input and the vector as the output. The course closes with linear transformations, which are functions that map an input vector   to an output vector. These various types of inputs and outputs also change in tandem. And technology allows us to explore and visualize this expanded genealogy of functions.

This is an exciting time to be a Precalculus teacher!

Greg Foley is the coauthor of the upcoming Demana/Waits AP® Precalculus text and MyLab course.

About the author: Dr. Greg Foley is the Morton Professor of Mathematics Education at Ohio University. Foley received BA and MA degrees in mathematics and a PhD in mathematics education from the University of Texas at Austin. He has taught arithmetic through graduate-level mathematics and coauthors Advanced Quantitative Reasoning: Mathematics for the World Around Us and Precalculus: Graphical, Numerical, Algebraic

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