Overview. Essential Questions. Precalculus, Quarter 1, Unit 1.4 Analyzing Exponential and Logarithmic Functions

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1 Analyzing Exponential and Logarithmic Functions Overview Number of instruction days: 5 7 (1 day = 53 minutes) Content to Be Learned Rewrite radical expressions using the properties of exponents. Rewrite exponential expressions in radical form. Identify the domain and range for exponential and logarithmic functions. Graph exponential functions, identifying intercepts and end behavior with and without the use of technology. Graph logarithmic functions, identifying intercepts and end behavior with and without the use of technology. Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Graph exponential and logarithmic functions, showing key features such as intercepts, asymptotes, domain, range, and end behavior. 2 Reason abstractly and quantitatively. Use the relationship between exponential and logarithmic functions to solve problems. 5 Use appropriate tools strategically. Use graphing calculators to graph exponential and logarithmic functions. Identify intercepts and describe the end behavior. Essential Questions What are the similarities and differences among exponential and logarithmic functions? What is the inverse relationship between logarithmic functions and exponential functions? How do you rewrite radicals using rational exponents and vice-versa? Why is it important to understand the opposite operation of all functions? Providence Public Schools D-25

2 Analyzing Exponential and Logarithmic Functions (5 7 days) Standards Common Core State Standards for Mathematical Content Number and Quantity The Real Number System N-RN Extend the properties of exponents to rational exponents. N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Functions Interpreting Functions F-IF Analyze functions using different representations F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Common Core State Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents D-26 Providence Public Schools

3 Analyzing Exponential and Logarithmic Functions (5 7 days) Precalculus, Quarter 1, Unit 1.4 and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Clarifying the Standards Prior Learning In Algebra I, students used rational exponents and their properties to rewrite expressions involving radicals. Students analyzed functions (linear and quadratic) using graphs, by hand, and using technology, showing intercepts, maxima, and minima. Students focused on linear and exponential functions, comparing their growth rate. In Algebra II, students graphed functions and expressed them symbolically, with and without technology. Current Learning In this unit, students work with more complex exponential functions. They also work with the properties of logarithmic functions. They graph exponential and logarithmic functions in a diversity of situations, including identifying intercepts, asymptotes, domain, range, and end behavior by hand and using graphing calculators in complex situations. Future Learning In AP Calculus AB, students will use rational exponents when calculating derivatives and antiderivatives. Exponential and logarithmic functions are used extensively throughout Calculus. Additional Findings Students sometimes struggle with the understanding of the inverse operations of exponential and logarithmic functions. In grades 9 12 all students should understand and compare the properties of classes of functions, including exponential [and] logarithmic... functions. Students should use technological tools to represent and study the behavior of polynomial, exponential, rational, and periodic functions, among others. (Principles and Standards for School Mathematics, pp ). Providence Public Schools D-27

4 Analyzing Exponential and Logarithmic Functions (5 7 days) Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Simplify and evaluate exponential and logarithmic expressions. Write logarithmic equations in exponential form. Write exponential equations in logarithmic form. Analyze domain and range for exponential and logarithmic functions. Graph exponential functions to describe the key characteristics, including intercepts and end behavior, with and without technology. Graph logarithmic functions to describe the key characteristics, including intercepts and end behavior, with and without technology. Learning Objectives Instruction Students will be able to: Simplify and evaluate expressions containing rational and irrational exponents. Use multiple representations to find key characteristics of graphs of exponential functions, including intercepts and end behavior. Simplify and evaluate expressions involving logarithms. Use multiple representations to find key characteristics of graphs of logarithmic functions, including intercepts and end behavior. Demonstrate understanding of concepts and skills related to exponential and logarithmic functions. D-28 Providence Public Schools

5 Analyzing Exponential and Logarithmic Functions (5 7 days) Precalculus, Quarter 1, Unit 1.4 Resources Advanced Mathematical Concepts: Precalculus with Applications, Glencoe, 2006, Teacher Edition and Student Edition Sections (pp ) Section 11-4 (pp ) TeacherWorks All-In-One Planner and Resource Center CD-ROM Exam View Assessment Suite Graphing Logarithms activity can be found at education.ti.com. See the Supplementary Materials section of this binder for the student and teacher notes for this activity. Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials TI-Nspire graphing calculators, graph paper, straight edge, gridded chart paper Instructional Considerations Key Vocabulary logarithmic function power function Planning for Effective Instructional Design and Delivery Reinforced vocabulary from previous grades or units: exponential function. Previously in PreCalculus, students graphed quadratic and polynomial functions and found the inverse of a function. Students will use this knowledge of the inverse of a function to rewrite exponentials in logarithmic form and vice versa. Graphing exponential and logarithmic functions should be of greatest emphasis as students will graph the functions with and without technology. Graphing opportunities are plentiful in the textbook and ancillary resources, but can also be extended by web research. The TI-Nspire graphing calculator activity, Graphing Logarithms, can be found on education.ti.com; the teacher and student notes are also included in the Supplementary Unit Materials section of this curriculum binder. As students locate additional examples, a visual display of their research adds to their ability to connect to the use of these functions. As time permits, use Jeopardy-type games or other game formats for student practice. When designed appropriately, game exercises can even be used as a quiz or formative-type assessment. Have students work in groups to identify similarities and differences when they compare exponential and logarithmic functions. Have them start with either an exponential or a logarithmic function and derive the inverse function. Then have students investigate, show in multiple representations, and present the graphs of each of the functions, connections of the inverse relationship, restrictions on the domains and ranges, y-intercepts, asymptotes, and other unique characteristics. In the presentation of their findings, students should compare and contrast the two functions. It is these characteristics that make each of these functions easy to recognize. Providence Public Schools D-29

6 Analyzing Exponential and Logarithmic Functions (5 7 days) Notes D-30 Providence Public Schools

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