Wentzville School District Algebra II

Transcription

1 Wentzville School District Algebra II Unit 6 - Rational Functions Unit Title: Rational Functions Course: Algebra II Brief Summary of Unit: In this unit, students will graph and analyze the properties of rational (reciprocal) functions. Students will simplify rational expressions and solve rational equations. Textbook Correlation: Glencoe Algebra II Chapter 8 Sections 1-4 and 6, however, section 6 will need to be supplemented with more problems similar to page 576: 1-4 and some rational proportions. Time Frame: 3-4 weeks WSD Overarching Essential Question Students will consider How do I use the language of math (i.e. symbols, words) to make sense of/solve a problem? How does the math I am learning in the classroom relate to the real-world? What does a good problem solver do? What should I do if I get stuck solving a problem? How do I effectively communicate about math with others in verbal form? In written form? How do I explain my thinking to others, in written form? In verbal form? How do I construct an effective (mathematical) argument? How reliable are predictions? Why are patterns important to discover, use, and generalize in math? How do I create a mathematical model? How do I decide which is the best mathematical tool to use to solve a problem? How do I effectively represent quantities and relationships through mathematical notation? How accurate do I need to be? WSD Overarching Enduring Understandings Students will understand that Mathematical skills and understandings are used to solve real-world problems. Problem solvers examine and critique arguments of others to determine validity. Mathematical models can be used to interpret and predict the behavior of real world phenomena. Recognizing the predictable patterns in mathematics allows the creation of functional relationships. Varieties of mathematical tools are used to analyze and solve problems and explore concepts. Estimating the answer to a problem helps predict and evaluate the reasonableness of a solution. Clear and precise notation and mathematical vocabulary enables effective communication and comprehension. Level of accuracy is determined based on the context/situation. Using prior knowledge of mathematical ideas can help discover more efficient problem solving strategies.

2 When is estimating the best solution to a problem? Concrete understandings in math lead to more abstract understanding of math. Transfer Students will be able to independently use their learning to know that operations and understandings developed in arithmetic can be extended to algebraic systems. know that rational functions and their graphs can be used to model real world phenomena. Meaning Essential Questions Understandings How do I know when a rational expression is simplified? What strategies would you use to solve a rational equation? Rational equations have solutions, some of which may be extraneous. Rational expressions can be rewritten in different forms to analyze their properties and make them more useful to perform operations upon. Why is a solution extraneous? Under what circumstances would you obtain an extraneous solution? How does the placement of the vertical asymptote correspond to the equation and the domain? How is a transformed function related to its parent function (reciprocal function)? Working with algebraic rational expressions and equations follows the same properties/rules as numerical rational expressions and equations. There are real world situations that can be modeled using rational functions. Horizontal asymptotes help to describe end behavior of a function. Rational expressions can be simplified in a more meaningful context. Transformations of functions can be an effective tool to graph and describe functions efficiently. Each component of a rational function provides

3 information about the corresponding graph, and vice versa. Acquisition Key Knowledge Key Skills Domain Simplify rational expressions extraneous solutions Rational expressions and equations reciprocal function horizontal asymptote Rewrite rational expressions (factoring, multiplying by a common denominator, etc.) so that two or more rational expressions can be added, subtracted, multiplied, divided, etc. Graph reciprocal functions using transformations with and without technology. vertical asymptote Interpret the meaning of the key features of the rational function from a graph or table in relation to the problem. (Key features may include intercepts, asymptotes (horizontal and vertical) and end behavior.) Use technology to discover the effects of parameter changes to reciprocal functions. Given a rational function, determine the practical domain of the function as it relates to the numerical relationship it describes as well as how it relates to its graph. Solve simple rational equations. Identify extraneous solutions. Solve real-world problems involving rational functions using multiple strategies. Standards Alignment

4 MISSOURI LEARNING STANDARDS A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. A.SSE.1 Interpret expressions that represent a quantity in terms of its context. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. Goal 1: 1, 4, 5, 6, 7, 8 Goal 2: 2, 3, 7 Goal 3: 1, 2, 3, 4, 5, 6, 7, 8 Goal 4: 1, 4, 5, 6 Show Me-Standards

5 Mathematics: 1, 4, 5