Wentzville School District Honors Algebra II

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1 Wentzville School District Honors Algebra II Unit Title: Radical Functions and Rational Exponents Course: Honors Algebra II Unit 4 - Radical Functions and Rational Exponents Brief Summary of Unit: In this unit, students will learn how to combine different functions together using function operations. Students will examine the properties of inverse functions and relations, and their relationship to the functions from which they came. Students will graph and analyze properties of square root and cube root functions. Students will solve radical equations, simplify and perform operations on radical expressions. In addition, students will be able to simplify and use expressions written in rational exponent form and radical form. Textbook Correlation: Glencoe Algebra 2 Chapter 6 ( , including Extended 6-4 lab) Time Frame: 3-4 weeks Note - Domain restrictions on radical functions should be emphasized throughout the unit in Honors Algebra II. 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 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 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? 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

2 How do I create a mathematical model? How do I decide which is the best mathematical tool to use to solve a How do I effectively represent quantities and relationships through mathematical notation? How accurate do I need to be? When is estimating the best solution to a 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. 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 math processes can be reversed methodically. Meaning Essential Questions Understandings What strategies would you use to solve a radical equation? Radical equations have solutions some of which may be extraneous. Why is a solution extraneous? Under what circumstances would you obtain an extraneous solution? How is a transformed function related to its parent function? When does a function have an inverse that is a function? What is the purpose of finding inverse functions? Transformations of functions can be an effective tool for graphing and describing functions efficiently. Inverses provide another way to analyze the related function. Complex functions are often the result of combining two or more simple functions. Key features of a function determine the type of function. How can simple functions be combined to form more complex functions? Different representations are more appropriate based on the application. What are real world examples where functions are combined? Functions can be represented in different but equivalent forms.

3 What are some real world examples where square functions are used? How can using and comparing different representations of similar types of functions lead to deeper understanding of key features? When is it best to graph by hand versus by using technology? How can you recognize the type of function given a variety of information? (graphically, algebraically, numerically in tables, or verbally) What is the best method for simplifying a radical expression? Acquisition Key Knowledge Key Skills Index Domain Extraneous solutions Effect of multiple parameter changes (vertical translations, horizontal translations, reflection over x-axis, vertical stretch) on radical functions. Inverse function notation Inverse functions and relations Function operations (adding, subtracting, multiplying, dividing and composition) Formal function notation. (f(x), (f+g)(x), f(x)+g(x), etc...) Composition notation (, ) Parent functions of square root and cube root nth root ( ) rationalizing the denominator Combine functions in order to represent more complex situations by adding, subtracting, multiplying, dividing, and composing. (This includes different function types.) Use function operations, including composition, to model real world situations. Find inverse functions. Use composition to verify functions are inverses. Given a square root function that includes multiple transformations, graph the function. Given a square root graph that has been transformed, write the function. (multiple transformations) Given the equation of a square root function, describe the effect of parameter changes. Use technology to graph and analyze radical functions. Graph square root, cube root functions and identify key features (intercepts, end behavior).

4 conjugate Compare features of functions represented in different ways (graphically, algebraically, numerically in tables, or verbally). For example, compare cube root functions where one is given as a table and the other is given as a function. Graph square root inequalities. Simplify radicals expressions Add, subtract, multiply, and divide radical expressions. o Rationalizing the denominator by multiplying by the conjugate should be emphasized o Expose Honors Algebra II students to simplifying nested radicals such as Write expressions with rational exponents in radical form, and vice versa. Simplify expressions in rational exponent form or radical form. Solve radical equations. Solve radical inequalities. Identify extraneous solutions Solve real-world world problems that involve square root functions and radical equations using multiple strategies. Standards Alignment MISSOURI LEARNING STANDARDS A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 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.

5 F.BF.4 Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2 x 3 or f(x) = (x+1)/(x-1) for x 1. b. (+) Verify by composition that one function is the inverse of another. F.BF.1 Write a function that describes a relationship between two quantities.* b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 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. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value 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 ). 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. Show Me-Standards 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 Mathematics: 1, 4, 5