- To verify by inspection, students can compare graphs of two relations and show that one is the reflection of the other across the line y = x.
- To verify by inspection, students can show that one table or set of coordinates is the inverse of another because the y-values of the first are the x-values of the second and vice-versa.
- To verify by inspection, students can show that a series of operations upon input values of one function are opposite and reversed in order for a second function.
- Students should be able to prove by composition that two functions are inverses of each other.
- To prove by composition means to determine if f(g(x)) = g(f(x)) = x.
- Students should be able to graph and identify key features of exponential and logarithmic functions, including domain, range, and x- and y-intercepts; roots, zeros, and solutions; asymptotes; interval(s) where the function is positive, and/or negative; non-symmetry; end behavior.
- Students should be able to calculate the average rate of change for a given interval, including the estimated rate of change.
- Students should have opportunities to gain an intuitive sense into what happens to the graph or model as a result of changes to the various key features of the function.
- Students should be given opportunities to solve real-life, culturally relevant problems involving the use of the common logarithm and the natural logarithm.
- Students should be able to apply their knowledge of the inverse relationship between exponential and logarithmic functions to solve real-life problems.
- Students should be able to solve problems involving exponential equations using the relationship with logarithmic functions to solve for the single unknown variable.
- Given pertinent information (e.g., ambient temperature and time), students should be able to use exponential equations to solve real-life problems and interpret the solutions.
- Students can solve and interpret equations that have one unknown variable, such as:
- Exponential growth
- Compound interest
- Given pertinent information, students should be able to use logarithmic equations to solve real-life problems and interpret the solutions.
- Students should be able to analyze what is happening in the relationships between quantities.
- Students should discuss the characteristics of exponential functions in context, including domain and range, zeros, intercepts, average rate of change, asymptote, and other relevant key features.
- Students should be able to solve real-life problems that can be modeled by exponential equations.
- Students should be encouraged to explore multiple solution pathways, which might include graphing with various tools, interpreting key features, and evaluating equations.
- Students can create, interpret and solve equations that have two unknown variables, such as:
- Half-Life
- Exponential growth
- Exponential decay
- Compound interest
- Students should be able to analyze and interpret logarithmic equations presented in mathematical, applicable situations.
- Students should discuss the characteristics of logarithmic functions in context, including domain and range, zeros, intercepts, average rate of change, asymptote, and other relevant key features.
- Students should be able to solve problems that can be modeled by logarithmic equations.
- Students should be encouraged to explore multiple solution pathways, which might include graphing with various tools, interpreting key features, and evaluating equations.
- Students are able to create and interpret equations involving logarithms such as the equation for the magnitude of an earthquakes M = log10 (I/S).
Textbook Connections
Module 7- Exponential Functions:
Lesson 1- Characteristics
Lesson 3- e
Module 8- Logarithmic Functions:
Lesson 1- Converting Forms
Lesson 2- Properties of Logarithms
Lesson 3- Common Logarithms
Lesson 4- Natural Logarithms
Lesson 5- Real World Exponentials and Logarithms
Module 7- Exponential Functions:
Lesson 1- Characteristics
Lesson 3- e
Module 8- Logarithmic Functions:
Lesson 1- Converting Forms
Lesson 2- Properties of Logarithms
Lesson 3- Common Logarithms
Lesson 4- Natural Logarithms
Lesson 5- Real World Exponentials and Logarithms