Math 3 Unit 5: Polynomial and Rational Representations and Modeling

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1 Approximate Time Frame: 5-6 weeks Connections to Previous Learning: In Math 1 and 2, students worked with one-variable equations. They studied linear, exponential and quadratic functions and compared them in a modeling context. Students looked at their multiple representations and how to translate them on a graph or in an equation. Students also looked at the structure of square root, cube root, piecewise, absolute value and step functions and represented these on graphs. Students manipulate expression and equations to reveal new information about these functions and how to solve systems with these types of functions. In previous units, students have worked with quadratic functions in various forms to determine their roots and vertices. They have used the factored form to determine if and where zeros exist. They have examined the equations of quadratic functions to determine the function's end behavior, and used all of this information to produce a graph of a quadratic. Focus of this Unit: Students will build on their knowledge of quadratics to include higher degree polynomial functions. They will use the factored form to determine zeros and the sign of the leading coefficient to determine end behavior. Multiplicity of factors will be used to help determine the behavior of the graph at its roots. These notions will all be extended to rational functions, which will initially be viewed as the reciprocal of a polynomial, and ultimately as the ratio of two polynomials. Students will see connections between these and the equations, functions, tables and graphs they have used in the past courses. Students will specifically compare polynomial and rational functions. Students will use these functions to create and compare models of real-life situations. Connections to Subsequent Learning: Students that continue their study of mathematics will continue to learn about new functions and more complex versions of the functions they have already studied. In college or a career situation, students will frequently be asked to model situations as they arise in real-life. Frequently, students will need to take data and represent it appropriately and make decisions from their representations. From the High School Algebra Progression Document, pp. 6-8: Polynomials form a rich ground for mathematical explorations that reveal relationships in the system of integers. For example, students can explore the sequence of squares 1, 4, 9, 16, 25, 36 And notice the difference between them 3, 5, 7, 9, 11 are consecutive odd integers. This mystery is explained by the polynomial identity. (n + 1) n n 2 = 2n + 1 A more complex identity, (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 allows students to generate Pythagorean triples. For example, taking x =1 and y = 2 in this identity yields 5 2 = Viewing polynomials as functions leads to explorations of a different nature. Polynomial functions are, on the one hand, very elementary, in that, unlike trigonometric and exponential functions, they are built up out of the basic operations of arithmetic. On the other hand, they turn out to be amazingly flexible, and can be used to approximate more advanced functions such as trigonometric and exponential functions. Although students only learn the complete story 11/17/2014 1:03:47 PM Adapted from UbD framework Page 1

2 here if and when they study calculus, experience with constructing polynomial functions satisfying given conditions is useful preparation not only for calculus, but for understanding the mathematics behind curve-fitting methods used in applications to statistics and computer graphics. A simple step in this direction is to construct polynomial functions with specified zeros. This is the first step in a progression which can lead, as an extension topic, to constructing polynomial functions whose graphs pass through any specified set of points in the plane. The analogy between polynomials and integers carries over to the idea of division with remainder. Just as in Grade 4 students find quotients and remainders of integers, in high school they find quotients and remainders of polynomials. The method of polynomial long division is analogous to, and simpler than, the method of integer long division. A particularly important application of polynomial division is the case where a polynomial p(x) is divided by a linear factor of the form x a for a real number a. In this case the remainder is the value p(a) of the polynomial at x = a. It is a pity to see this topic reduced to synthetic division, which reduced the method to a matter of carrying numbers between registers, something easily done by a computer, while obscuring the reasoning that makes the result evident. It is important to regard the Remainder Theorem as a theorem, not a technique. A consequence of the Remainder Theorem is to establish the equivalence between linear factors and zeros that is the basis of much work with polynomials in high school: the fact that p(a) = 0 if and only if x a is a factor of p(x). It is easy to see if x a is a factor then p(a) = 0. But the Remainder Theorem tells us that we can write p(x) = (x a)q(x) + p(a) for some polynomial q(x). In particular, if p(a) = 0 then) (x) = (x a)q(x), so x a is a factor of p(x). Creating Equations Students have been writing equations, mostly linear equations, since middle grades. At first glance it might seem that the progression from middle grades to high schools is fairly straightforward: the repertoire of functions that is acquired during high school allows students to create more complex equations, including equations arising from linear and quadratic functions, and simple rational and exponential functions; students are no longer limited largely to linear equations in modeling relationships between quantities with equations in two variables. 11/17/2014 1:03:47 PM Adapted from UbD framework Page 2

3 Desired Outcomes Standard(s): Interpret the structure of expressions 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 ). Create equations that describe numbers or relationships. A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Understand solving equations as a process of reasoning and explain the reasoning. A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Reason quantitatively and use units to solve problems. N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. Interpret functions that arise in applications in terms of the context 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 function in increasing, decreasing, positive, or negative, relative maximums and minimums, symmetries, end behavior, and periodicity. F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Analyze functions using different representations. F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand the relationship between zeros and factors of polynomials. A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x) A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. 11/17/2014 1:03:47 PM Adapted from UbD framework Page 3

4 Represent and solve equations and inequalities graphically. 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 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. Use polynomial identities to solve problems. A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example: the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. Rewrite rational expressions 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. Build new functions from existing functions. 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. F.BF.4a 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. Transfer: Students will apply concepts and procedures regarding solving polynomial and rational equations and the relationship between zeros and factors of polynomials in order to justify a method of solution. Example: Given a radical equation, students will solve and explain the existence of extraneous solutions. Example: Given x 3 + x 2 ax 9 and the fact that x 3 is a factor of the polynomial, students will determine the value of a. Students will apply concepts and procedures regarding a relationship between two quantities in order to model situations with polynomial and rational functions. Example: Given a scenario, students will define quantities, create a polynomial or rational equation, obtain a result, and interpret and validate the result. 11/17/2014 1:03:47 PM Adapted from UbD framework Page 4

5 Understandings: Students will understand A polynomial function, in the variable x, may be written as Math 3 Unit 5: Polynomial and Rational Representations and Modeling p( x) a x a x a x a x a x a x, with a n 0, n n1 n n n1 n where n is a non-negative integer (called the degree of the polynomial) and each coefficient, ai is a real number for i = 0, 1,..., n. Otherwise, the polynomial may be equal to 0, and in this case, we say the degree is undefined. The degree of a polynomial helps to determine the end behavior of its graph The zeros of each factor of a polynomial determine the x-intercepts of its graph The Remainder Theorem can be used to determine roots of polynomials The long division algorithm for polynomials can be used to rewrite rational expressions in different forms. Rational equations can be created from given information and solved algebraically or graphically. Nonlinear systems can be solved graphically, algebraically or from a table. Graphs of rational functions are often discontinuous, due to values that are not in the domain of the function. Essential Questions: What is a polynomial? What is a rational function? How does the degree of a polynomial affect its graph? How do the factors of a polynomial affect its graph? How can you create and solve equations involving rational equations? When changes are made to an equation, what changes are made to the graph? What are the different methods that can be used to find the solutions of a system of equations? 11/17/2014 1:03:47 PM Adapted from UbD framework Page 5

6 Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) 1. Make sense of problems and persevere in solving them. * 2. Reason abstractly and quantitatively. Use factors to determine zeros of polynomials, leading coefficients to determine direction of ends of the graph of a polynomial. Also, use zeros of the polynomial in the denominator of a rational function to determine location of vertical asymptotes, and compare degrees of numerator and denominator to determine horizontal asymptotes. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. Students will use graphs, tables, and equations to model polynomial and rational equations. Students will interpret the * appropriateness of linear, exponential, quadratic, polynomial or rational models for a given data set or situation. 5. Use appropriate tools strategically. Use a computer algebra system to factor polynomials; use graphing technology to explore graphs of various polynomials and rational functions; use technology (graphs, tables) to solve for a point of intersection of two functions. * 6. Attend to precision. Graph a polynomial or rational function while carefully indicating zeros, asymptotes, y-intercept, and other significant features. * 7. Look for and make use of structure. Recognize that a quadratic function with a repeated factor does not intersect the x-axis in two unique points, and extend this notion to write equations of polynomials or rational functions which have graphs that do not pass through the x-axis at the root of a factor having even multiplicity. 8. Look for and express regularity in repeated reasoning. 11/17/2014 1:03:47 PM Adapted from UbD framework Page 6

7 Prerequisite Skills/Concepts: Students should already be able to: Factor a quadratic expression over the integers. Determine zeros from a factored form of a quadratic. Manipulate a quadratic function between various forms to determine key features, including zeros, the vertex, maximum/minimum value, and end behavior. Represent and solve one variable equations and know the steps for solving equations and looking at equivalent forms limited to linear, exponential and quadratic equations. Choose and interpret appropriate models limited to linear, exponential and quadratic functions. Define a function and the different parts in expressions, equations and functions. Recognize how functions can be represented on a graph and in a table and how scale and labels can change the look of a representation. Interpret these representations with regards to linear, exponential and quadratic functions. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Estimate, calculate and interpret average rate of change over a specified interval. Identify the effect on the graph of replacing f(x)by f(x) + k, kf(x), f(kx), and f(x + k). Advanced Skills/Concepts: Some students may be ready to: Find the conjugate of a complex number and use them to find quotients of complex numbers or solve for complex roots of a polynomial. Determine the magnitude of a complex number. Graph a complex number. Calculate complex roots of functions. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication and division by a nonzero rational expression; add, subtract, multiply and divide rational expressions. Determine whether a rational function has a horizontal or oblique asymptote. Calculate the equation of a horizontal or oblique asymptote of a rational function. Determine if and where a rational function crosses its horizontal (or oblique) asymptote. F.IF.7 (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Write equations of rational functions, given the graph. Roughly graph the reciprocal function of a given graph (given polynomial, reciprocal of polynomial, or ratio of two polynomials). Use advanced skills of the complex numbers before, could find the conjugate of a complex number and use conjugates to find moduli and quotients of complex number when working with rational equations and functions. (N.CN.3+) A.APR.7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. F.BF.4 Find Inverse functions. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain. 11/17/2014 1:03:47 PM Adapted from UbD framework Page 7

8 Knowledge: Students will know The Remainder Theorem and how it applies to determining whether a number is a root of the polynomial. Skills: Students will be able to Prove and use polynomial identities Graph a polynomial given in factored form, indicating all intercepts and directions of end behaviors. Use the structure of an expression to identify ways to rewrite it. Create equations in one, two, or more variables and use them to solve problems. Construct viable arguments to justify a solution method. Write the equation of a polynomial function given its graph or defining characteristics of its graph. Solve simple rational equations in one variable, checking for extraneous solutions. Perform the long division algorithm for polynomials in order to rewrite simple rational expressions in different forms Use technology (graphs, tables) to solve the equation f(x) = g(x), where f(x) and/or g(x) are polynomial or rational functions. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. 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. Solve an equation of the form f(x) = c for a simple function f that has an inverse. Write an expression for the inverse for a simple function. For example, f(x) = (x+1)/ (x-1) for x 1. Write a polynomial equation and/or function to model a real-life situation. Use a model of a polynomial function to interpret information about a real-life situation. Define appropriate quantities for the purpose of descriptive modeling. Translate among representations of polynomial functions including tables, graphs, equations and real-life situations. 11/17/2014 1:03:47 PM Adapted from UbD framework Page 8

9 Distinguish between linear, exponential, quadratic, polynomial and rational functions from multiple representations. Rewrite polynomial and rational equations to reveal new information. Compare two functions represented in different ways. Create a scatterplot from data and interpret the relationship of the quantities represented. 11/17/2014 1:03:47 PM Adapted from UbD framework Page 9

10 WIDA Standard: (English Language Learners) English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. English language learners benefit from: Explicit vocabulary instruction with regard to the features and types of equations, functions, and graphs. Guided discussion regarding the connections between graphs, tables, equations, expressions and the contexts they represent. Academic Vocabulary: Critical Terms: Polynomial Degree of term Degree of polynomial Factor of polynomial Root/zero/x-intercept Multiplicity of root Rational function Supplemental Terms: integer term coefficient2) domain range discontinuity Tables Graphs Real-life situations Equations 11/17/2014 1:03:47 PM Adapted from UbD framework Page 10

11 Assessment Pre-Assessment Formative Assessments Summative Assessments Self-Assessments Self Assessment Washing Machine Dilemma Blood Alcohol Content Building Rational Functions Rational Graphs Sort Unit 4 Summative Assessment Sample Lesson Sequence 1. Working with polynomial expressions (division) and identities A.APR.4, A.APR.6, A.SSE.2, A.APR.2 2. Graph and analyze polynomial functions. A.CED.2, F.IF.9, A.APR.3, F.BF.4 3. Polynomials in the Real World F.IF.4, F.IF.6, F.IF.7c 4. Washing Machine Dilemma (A.CED.1, A.REI.11), Blood Alcohol Content (F.IF.4, F.IF,9, F.BF.3), Building Rational Functions (A.CED.1-2), Rational Function Sort (F.IF.4, A.APR.6, F.BF.3) 5. Creating and solving simple (deg 2) rational equations. A.SSE.2, A.APR.6, A.REI /17/2014 1:03:47 PM Adapted from UbD framework Page 11