# Unit 2Linear and Exponential Functions

## Lesson 1

### Learning Focus

Represent situations with different types of growth.

Compare models for situations that occur over time.

### Lesson Summary

In this lesson, we learned that the possible inputs for a function are called the domain. We found that some situations are best described using a discrete model and others are represented better with a continuous model. Arithmetic sequences are part of the linear family of functions and geometric sequences are part of the exponential family of functions.

## Lesson 2

### Learning Focus

Use representations to model situations with linear and exponential functions.

Determine when a discrete model or continuous model is most appropriate.

### Lesson Summary

In this lesson, we modeled linear and exponential functions and learned to identify features that allow us to determine whether a discrete or a continuous model is more appropriate. We discussed number sets and used them to write function domains using set builder notation.

## Lesson 3

### Learning Focus

Use representations to determine if a function is discrete or continuous.

Determine the domain of a function.

### Lesson Summary

In this lesson, we used the definitions of linear and exponential functions to determine if functions were linear or exponential. We learned to identify equal intervals on a continuous function so that we could tell if there were equal differences or change factors. We practiced determining domains and whether a function is discrete or continuous.

## Lesson 4

### Learning Focus

Explore the meaning of a fraction as an exponent.

### Lesson Summary

In this lesson, we developed meaning for using a fraction as an exponent. We began with a geometric sequence and found a way to fit data at the points halfway between whole number inputs in a way that would maintain the multiplicative behavior of the sequence. We continued to work with different fractional inputs and gave meaning to using a variety of fractions as exponents.

## Lesson 5

### Learning Focus

Relate the key features of exponential functions to properties of negative exponents.

Rewrite exponential expressions that involve negative exponents.

### Lesson Summary

In this lesson, we noticed several characteristics of the graphs and tables of exponential functions that can be explained using our understanding of negative exponents. We also used the rules of exponents to change the form of numeric expressions that contain negative exponents.

## Lesson 6

### Learning Focus

Examine how the properties of exponents work with rational exponents.

Write equivalent exponential functions using different growth factors.

### Lesson Summary

In this lesson, we continued to explore the meaning of rational exponents, including negative integer exponents and fractional exponents. We learned that the properties of exponents can be applied to all rational exponents, not just integer exponents.

## Lesson 7

### Learning Focus

Change the form of radical expressions using properties of exponents.

### Lesson Summary

In this lesson, we learned how to change the form of complicated radical and exponential expressions using the properties of radicals and exponents. Strategies for changing the form of radical expressions can be explained by converting the radical expressions to exponential form.

## Lesson 8

### Learning Focus

Interpret mathematical models to make business decisions.

Determine which type of function grows faster and make arguments about why.

### Lesson Summary

In this lesson, we modeled the growth of two businesses and made comparisons. We used our representation to find when the two businesses had the same net income and to justify which business was the best investment. We found that exponential functions exceed linear functions for large values of and that the point of intersection of the two functions has meaning in a realistic context.

## Lesson 9

### Learning Focus

Understand and find the average rate of change of a function in an interval.

Develop a formula for the average rate of change for any function.

### Lesson Summary

In this lesson, we learned to find the average rate of change of a function in an interval. We learned that the average rate of change is calculated by finding the change in and dividing by the change in . Sometimes an equation is used to calculate the -values at the beginning and end of the interval, and sometimes a graph is used to find the heights at the beginning and end of the interval. In either cases, the -values are subtracted and then that amount is divided by the width of the interval, or the difference between the -values.

## Lesson 10

### Learning Focus

Find patterns that are useful in writing equations for linear functions.

### Lesson Summary

In this lesson, we learned a new and efficient pattern for writing the equation of a line. The method can be used with a table, a graph, or any two points on the line.

## Lesson 11

### Learning Focus

Use different forms of linear and exponential functions to efficiently write equations.

Use the information given in different forms of equations to graph functions.

### Lesson Summary

In this lesson, we summarized our work with writing equations for linear and exponential functions. We worked on strategically selecting a useful form for the context by identifying the information about the type of change and the initial values, wherever they are.