Algebra 2 Unit 4: Exponential and Logarithmic Functions

Unit Driving Question

•  How can exponential and logarithmic functions be used to represent real world scenarios?

Essential Questions 

1. What are the distinguishing characteristics of an exponential function and logarithmic function?

2. How can we identify key features of a exponential and logarithmic functions?

3. What mathematical and real world value do exponential and logarithmic functions possess?

4. How are exponential and  logarithmic functions used to solve real world problems?

5. How are sequences and series related to functions?

Launch Task 

Big Ideas for Development Lessons 

Closure & Assessment 

1 Lesson 

8 Weeks (approximately 1-2 weeks per big idea) 

1 Week 

• Multiplying Cells

Click on the links below to see each Big Idea's Lesson Overview (includes links to teacher notes and student activities)

1. Exponential functions are defined as f(x)=a^x.
2. Exponential functions share similar characteristics with power functions.
3. Logarithms and exponents can be used to solve mathematical and real-world problems.
4. Sequences  can be described as functions. Sums of series can be represented by exponential and logarithmic functions. 

• Dead Cell Phone Performance Assessment (Covers A2.A.1.2, A2.A.1.6, A2.A.1.7, and A2.F.2.4)- For access to scoring rubric and student samples, complete the NextThought PLC Common Assessment Discussion, Implementation, and Analysis training module.  

Big Idea 1: Exponential functions are defined as f(x)=a^x

OAS-M: A2.A.1.2, A2.F.1.4, A2.D.1.2

Lessons and Additional Activities

Big Idea 1 Lessons 1-4 Overview (includes links to teacher notes and student activities)

Additional Activities

• XBOX Xponential (Mathalicious) Students create exponential models to predict the speed of video game processors over time, compare their predictions to observed speeds, and consider the consequences as digital simulations become increasingly lifelike.

Evidence of Understanding 

Discover and describe characteristics of exponential functions in mathematical and real world situations.

• Identify the critical points and key characteristics of the an exponential function using technology (ie. desmos or graphing calculator).

• Compare and contrast growth and decay functions using technology.

• Conjecture how growth and decay functions are similar and different algebraically. Verify results using technology using a table or graph.

• Explore real world situations requiring the use of a growth or decay function displayed in a variety of ways (ie. graph, table, scatter plot, sequence, etc.) . Include generalizations about vocabulary and patterns.

• Justify and recognize the need for a growth or decay function given a graph, table, scatter plot, sequence or scenario.

• Research real world applications. Identify the common bases used.

• Build a growth or decay function given at least 3 points. Justify your function rule using technology.

Build exponential functions using information from a graph, table or situation.

• Discover how an exponential function can be related to a recursive function.

• Identify key components of an exponential function. (i.e. initial value, fixed proportion of change, growth or decay, etc.)

• Build exponential equations, graphs or tables given a situation.

• Recognize how the input and output are related on the graph, table and equations of exponential function. Describe the meaning in various situations.

Big Idea 2: Exponential functions share similar characteristics with power functions.

A2.A.1.7, 

A2.F.1.4, 

A2.F.2.4

Lessons and Additional Activities

Big Idea 2 Lessons 1-4 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

Compare and contrast the graphs of power and exponential functions.

• Recognize differences and similarities of the graphs of power and exponential functions.

• Identify appropriate situations to use exponential growth or decay functions, power functions, or none of these types given a graph, table, data points, sequence or real world scenario.

Compare and contrast the function rules of power and exponential functions.

• Conjecture about the relationship between using laws of exponents with power functions and exponential functions.

• Explore equivalent exponential functions using technology (Desmos or graphing calculator). Justify the equivalency.

• Recognize when to use equivalent bases to solve exponential equations.

• Solve exponential equations using technology (Desmos or graphing calculator) and algebraically. 

Big Idea 3: Logarithms and exponents can be used to solve mathematical and real-world problems.

OAS-M: A2.A.1.2, A2.A.1.8, A2.F.2.4 

Lessons and Additional Activities

Big Idea 3 Lessons 1-4 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

Discover and describe the relationship between an exponential function and a logarithmic function.

• Compare and contrast the graphs of an exponential function and logarithmic function with the same base.

• Identify any similarities, differences or patterns found when investigating critical points and key characteristics of the graph.

• Create a logarithmic function when given an exponential function and vice versa. Justify that logarithmic functions and exponential functions are inverses of each other algebraically and graphically (ex. Use Desmos or graphing calculator.)

Recognize the necessity of converting an exponential function to logarithmic and vice versa when given a mathematical or real world situation.

• Justify converting between exponential and logarithmic functions when solving for an unknown quantity.

• Solve logarithmic equations using technology (desmos or graphing calculator) and algebraically.

• Explore situations using technology that justify the equivalence of converting exponential and logarithmic functions.

• Recognize restrictions or unnecessary domain and range values in mathematical and real world scenarios.

• Identify the representation of input and output values for various scenarios.

• Analyze the exponential or logarithmic model for the usefulness of the graph in its entirety for the given scenario. Use technology such as desmos or a graphing calculator.

• Does the graph model the situation without end, or is a domain restriction necessary/useful?

Big Idea 4: Sequences  can be described as functions. Sums of series can be represented by exponential and logarithmic functions

OAS-M:  A2.A.1.7, A2.D.1.2

Lessons and Additional Activities

Big Idea 4 Lessons 1-4 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

Discover patterns in sequences and identify as arithmetic, geometric or neither.

• Identify sequences that can be written as a generalization.

• Notice that arithmetic sequences are formed using a common difference.

• Notice that geometric sequences are formed using a common ratio.

Discover and make generalizations about graphs of arithmetic and geometric sequences.

• Identify key characteristics of the graphs using technology.

• Compare and contrast the graphs and function rules of arithmetic and geometric.

Build explicit and recursive formulas for geometric and arithmetic sequences.

• Identify from a table, data set, graph or scenario as arithmetic or geometric sequence.

• Develop a rule to represent the data.

• Research real world applications where explicit and recursive formulas are used.

• Compare and contrast the graphs of explicit and recursive formulas. Identify when and how to use each.

Solve real-world and mathematical problems that can be modeled using arithmetic or finite geometric sequences or series given the nth terms and sum formulas.

• Identify a sum formula as arithmetic or geometric.

• Identify when and how to use both arithmetic and geometric sum formulas.

• Recognize summation notation and identify key parts of the notation.