Unit 2 Linear and Exponential Functions
Represent situations with different types of growth.
Compare models for situations that occur over time.
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.
Use representations to model situations with linear and exponential functions.
Determine when a discrete model or continuous model is most appropriate.
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.
Use representations to determine if a function is discrete or continuous.
Determine the domain of a function.
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.
Relate the key features of exponential functions to properties of negative exponents.
Rewrite exponential expressions that involve negative exponents.
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.
Change the form of algebraic expressions using properties of exponents.
In this lesson, we learned how to change the form of complicated exponential expressions using the properties of exponents. It is often useful and conventional to write algebraic expressions without negative exponents, which can be done by applying the definition of a negative exponent.
Make modeling decisions about business plans.
Interpret mathematical models to make business decisions.
Determine which type of function grows faster and make arguments about why.
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
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.
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
Find patterns that are useful in writing equations for linear functions.
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.
Use different forms of linear and exponential functions to efficiently write equations.
Use the information given in different forms of equations to graph functions.
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.