Algebra 2 (& GT)
Unit 6: Radical and Rational Functions
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What will my child learn?
Students will:
Analyze functions using different representations.
F.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases (SAT® Content  PAM.13).
b. Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions.
Extend the properties of exponents to rational exponents.
N.RN.A.1
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)^3 = 5^(1/3)3 to hold so (5^1/3)^3 must equal 5.
N.RN.A.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents (SAT® Content  PAM.07).
Understand solving equations as a process of reasoning and explain the reasoning.
A.REI.A.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise (SAT® Content  PAM.08).
Analyze functions using different representations.
F.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
Understand the relationship between zeros and factors of polynomials.
A.APR.B.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).
Rewrite rational expressions.
A.APR.D.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 (SAT® Content  PAM.07).
Represent and solve equations graphically.
A.REI.D.11
Explain why the xcoordinates 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, and rational functions.*
Students will:
Analyze functions using different representations.
F.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases (SAT® Content  PAM.13).
b. Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions.
 Check for Understanding: Graphs of Absolute Value Functions  Graphs of Piecewise Linear Functions
 Review/Rewind: Intro to Graphs of Absolute Value Functions  Intro to Piecewise Functions
 Enrichment Tasks: Bank Account Balance
Extend the properties of exponents to rational exponents.
N.RN.A.1
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)^3 = 5^(1/3)3 to hold so (5^1/3)^3 must equal 5.
 Background Info.
 Check for Understanding: Understanding Fractional Exponents
 Review/Rewind: Introduction to Rational Exponents
 Enrichment Tasks: Extending the Definitions of Exponents, Variation 2  Evaluations Exponential Expressions
N.RN.A.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents (SAT® Content  PAM.07).
 Background Info.
 Check for Understanding: Adding and Subtracting Radicals  Fractional Exponents
 Review/Rewind: Simplifying Rational Exponents
 Enrichment Tasks: Kelper's Third Law of Motion  Rational or Irrational?
Understand solving equations as a process of reasoning and explain the reasoning.
A.REI.A.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise (SAT® Content  PAM.08).
 Background Info.
 Check for Understanding: Extraneous Solutions to Radical Equations  Extraneous Solutions to Rational Equations
 Review/Rewind: Extraneous Solutions to Radical Equations Review  Equation with Two Rational Expressions
Analyze functions using different representations.
F.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
 Check for Understanding: Graphs of Rational Functions
 Review/Rewind: Graphs of Rational Functions Review
Understand the relationship between zeros and factors of polynomials.
A.APR.B.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).
 Background Info.
 Check for Understanding: Reminder Theorem of Polynomials
 Review/Rewind: Intro to the Polynomial Remainder Theorem
 Enrichment Tasks: Zeros and Factorization of a Non Polynomial Function  The Missing Coefficient
Rewrite rational expressions.
A.APR.D.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 (SAT® Content  PAM.07).
 Background Info.
 Check for Understanding: Dividing Polynomials with Remainders
 Review/Rewind: Practice Dividing Polynomials with Remainders
 Enrichment Tasks: Combined Fuel Efficiency  Egyptian Fractions II
Represent and solve equations graphically.
A.REI.D.11
Explain why the xcoordinates 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, and rational functions.*
 Background Info.
 Check for Understanding: Intersecting Functions  Systems of Nonlinear Equations
 Review/Rewind: Solving Equations Graphically
 Enrichment Tasks: Introduction to Polynomials  College Fund  Ideal Gas Law
What are some signs of student mastery?

Tools & Technology
Desmos is a free online graphing calculator that works on any computer or tablet without requiring any downloads. A FREE Desmos iPad app is available too! See the steps for simplifying radical expressions using the free calculator by Symbolab. 