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average rates of change to more precisely discuss how quantities are changing and to describe how fast graphs are increasing and decreasing.
During the later part of this unit, students learn about inverse functions using simple encryption methods. Encoding and decoding messages requires you to apply a rule and then to reverse the steps to decode them. Similarly, a function rule can be undone by reversing the steps of the function. The process can be described with an inverse function.
Throughout this unit, students use and connect the diWerent representations of functions, tables, graphs, equations, and verbal descriptions with mathematical modeling activities (MP4). The culminating lesson of this unit combines many of the function ideas in two modeling activities: predicting how long it will take for a cell phone battery to charge and to lose its charge given incomplete data.
In subsequent units and courses, students will continue use what they learned in this unit to learn about new types of functions, for example, quadratic, exponential, logarithmic, and periodic functions. They will also model real-world situations characterized by these new types of functions.
F3 Introduction to Exponential Functions
Before starting this unit, students are familiar with linear functions from previous units in this course and from work in grade 8. They have been formally introduced to functions and function notation and have explored the behaviors and traits of both linear and non-linear functions. Additionally, students have spent signiTcant time graphing, interpreting graphs, and exploring how to compare the graphs of two linear functions to each other. In this unit, students frequently use the properties of exponents, a topic developed in grade 8. They also apply their understanding of percent change from grade 7 and use an exponent to express repeated increase or decrease by the same percentage.
In this unit, students are introduced to exponential relationships. Students learn that exponential relationships are characterized by a constant quotient over equal intervals, and compare it to linear relationships which are characterized by a constant diWerence over equal intervals. They encounter contexts that change exponentially. These contexts are presented verbally and with tables and graphs. They construct equations and use them to model situations and solve problems.
Students view these new types of relationships as functions and employ the notation and terminology of functions (for example, dependent and independent variables). They study graphs of exponential functions both in terms of contexts they represent and abstract functions that don't represent a particular context, observing the eWect of diWerent values
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