Unit Summary
In Unit 4, students will extend their understanding of inverse functions to functions with a degree higher than 1. Alongside this concept, students will factor and simplify rational expressions and functions to reveal domain restrictions and asymptotes. Students will become fluent in operating with rational and radical expressions and use the structure to model contextual situations. In this unit, students will also revisit the concept of an extraneous solution, first introduced in Unit 1, through the solution of radical and rational equations.
The unit begins with Topic A, where there is a focus on understanding the graphical and algebraic connections between rational and radical expressions, as well as fluently writing these expressions in different forms. In Topic B, students delve deeper into rational equations and functions and identify characteristics such as the $$x$$- and $$y$$-intercepts, asymptotes, and removable discontinuities based on the relationship between the degree of the numerator and denominator of the rational expression. Students will also connect these features with the transformation of the parent function of a rational function. In Topic C, students solve rational and radical equations, identifying extraneous solutions, then modeling and solving equations in situations where rational and radical functions are necessary. Students will connect the domain algebraically with the context and interpret solutions.
Pacing: 20 instructional days (18lessons, 1 flex day, 1 assessment day)
Assessment
The following assessments accompany Unit 4.
Post-Unit
Use the resources below to assess student understanding of the unit content and action plan for future units.
Post-Unit Assessment
Unit Prep
Intellectual Prep
Suggestions for how to prepare to teach this unit
Internalization of Standards via the Unit Assessment
- Take unit assessment. Annotate for:
- Standards that each question aligns to
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) that assessment points to
Internalization of Trajectory of Unit
- Read and annotate “Unit Summary."
- Notice the progression of concepts through the unit using “Unit at a Glance.”
- Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions
Essential Understandings
The central mathematical concepts that students will come to understand in this unit
- A rational function is a ratio of polynomial functions. If a rational function does not have a constant in the denominator, the graph of the rational function features asymptotic behavior and can have other features of discontinuity.
- Rational and radical equations that have algebraic numerators or denominators operate within the same rules as fractions.
- Extraneous solutions may result due to domain restrictions in rational or radical functions.
- Rational functions can be used to model situations in which two polynomials or root functions are divided.
Vocabulary
Terms and notation that students learn or use in the unit
| Vertical and horizontal asymptote | Invertible functions |
| Rational function | Zero product property |
| Rational expression | Asymptotic discontinuities (infinite) |
| Domain restriction | Removable discontinuities |
| Square root / cube root | End behavior |
| Extraneous solutions | Sign chart |
Lesson Map
Topic A: Introduction to Rational and Radical Functions and Expressions
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Topic B: Features of Rational Functions and Graphing Rational Functions
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Topic C: Solve Rational and Radical Equations and Model with Rational Functions
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Common Core Standards
Key
Major Cluster
Supporting Cluster
Additional Cluster
Core Standards
The content standards covered in this unit
Arithmetic with Polynomials and 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.
Arithmetic with Polynomials and Rational Expressions
A.APR.D.6— Rewrite simple rational expressions in different forms; write <sup>a(x </sup>/<sub>b(x)</sub> in the form q(x) + <sup>r(x)</sup>/<sub>b(x)</sub>, 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.APR.D.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.
Arithmetic with Polynomials and Rational Expressions
A.APR.D.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.
Building Functions
F.BF.B.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.
Building Functions
F.BF.B.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.B.4— Find inverse functions.
Building Functions
F.BF.B.4— Find inverse functions.
Creating Equations
A.CED.A.2— Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Creating Equations
A.CED.A.2— Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
High School — Number and Quantity
N.Q.A.1— Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
High School — Number and Quantity
N.Q.A.1— Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
N.RN.A.2— Rewrite expressions involving radicals and rational exponents using the properties of exponents.
High School — Number and Quantity
N.RN.A.2— Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Interpreting Functions
F.IF.B.5— Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Interpreting Functions
F.IF.B.5— Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
F.IF.C.7.B— Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
Interpreting Functions
F.IF.C.7.B— Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F.IF.C.7.D— Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
Interpreting Functions
F.IF.C.7.D— Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
Reasoning with Equations and Inequalities
A.REI.A.2— Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Reasoning with Equations and Inequalities
A.REI.A.2— Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A.REI.D.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.Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Reasoning with Equations and Inequalities
A.REI.D.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.Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Foundational Standards
Standards covered in previous units or grades that are important background for the current unit
Arithmetic with Polynomials and Rational Expressions
A.APR.A.1
Arithmetic with Polynomials and Rational Expressions
A.APR.A.1— Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Building Functions
F.BF.B.3
Building Functions
F.BF.B.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.B.4.A
Building Functions
F.BF.B.4.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 x3 or f(x) = (x+1)/(x—1) for x ? 1.
Creating Equations
A.CED.A.4
Creating Equations
A.CED.A.4— Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.For example, rearrange Ohm's law V = IR to highlight resistance R.
Expressions and Equations
8.EE.A.1
Expressions and Equations
8.EE.A.1— Know and apply the properties of integer exponents to generate equivalent numerical expressions.For example, 3² × 3<sup>-5</sup> = 3<sup>-3</sup> = 1/3³ = 1/27.
Interpreting Functions
F.IF.A.1
Interpreting Functions
F.IF.A.1— Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F.IF.B.4
Interpreting Functions
F.IF.B.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.Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
F.IF.C.8
Interpreting Functions
F.IF.C.8— Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F.IF.C.8.A
Interpreting Functions
F.IF.C.8.A— Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Reasoning with Equations and Inequalities
A.REI.A.1
Reasoning with Equations and Inequalities
A.REI.A.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.
Seeing Structure in Expressions
A.SSE.A.1
Seeing Structure in Expressions
A.SSE.A.1— Interpret expressions that represent a quantity in terms of its contextModeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Standards for Mathematical Practice
CCSS.MATH.PRACTICE.MP1— Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2— Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3— Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4— Model with mathematics.
CCSS.MATH.PRACTICE.MP5— Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6— Attend to precision.
CCSS.MATH.PRACTICE.MP7— Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8— Look for and express regularity in repeated reasoning.
Unit 3
Polynomials
Unit 5
Exponential Modeling and Logarithms
