# Exponential functions, rates of change, and the multiplicative unit

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## Abstract

Conventional treatments of functions start by building a rule of correspondence between*x*-values and*y*-values, typically by creating an equation of the form*y=f*(*x*). We call this a*correspondence* approach to functions. However, in our work with students we have found that a*covariational* appraoch is often more powerful, where students working in a problem situation first fill down a table column with*x*-values, typically by adding 1, then fill down a*y*-column through an operation they construct within the problem context. Such an approach has the benefit of emphasizing rate-of-change. It also raises the question of what it is that we want to cal ‘rate’ across different functional situations. We make two initial conjectures, first that a rate can be initially understood as a*unit per unit* comparison and second that a unit is the*invariant relationship between a successor and its predecessor*. Based on these conjectures we describe a variety of multiplicative units, then propose three ways of understanding rate of change in relation to exponential functions. Finally we argue that rate is different than ratio and that an integrated understanding of rate is built from multiple concepts.

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