Students will extend their understanding of inverse functions to functions with a degree higher than 1, and factor and simplify rational expressions to reveal domain restrictions and asymptotes.

## Unit Summary

In Unit 4, Rational and Radical Functions, 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 (18 lessons, 1 flex day, 1 assessment day)

## Assessment

This assessment accompanies Unit 4 and should be given on the suggested assessment day or after completing the unit.

## Unit Prep

### Intellectual Prep

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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.”
• Essential understandings
• Connection to assessment questions

### Essential Understandings

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• 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

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 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

# 1

A.APR.D.6

F.IF.B.5

Define rational functions. Identify domain restrictions of rational functions.

# 2

F.BF.B.4

F.IF.C.7.B

Identify domain restrictions algebraically for non-invertible functions.

# 3

F.BF.B.3

F.IF.C.7.B

Graph and transform square root and cubic root functions.

# 4

F.IF.B.5

N.RN.A.2

Write rational functions in equivalent radical form and identify domain restrictions of rational and radical functions.

# 5

N.RN.A.2

Write radical and rational exponent expressions in equivalent forms.

# 6

A.APR.D.6

A.APR.D.7

Multiply and divide rational expressions and simplify using equivalent expressions.

A.APR.D.6

A.APR.D.7

# 8

A.APR.D.6

F.IF.C.7.D

Identify asymptotic discontinuities (also known as infinite discontinuities) and removable discontinuities in a rational function and describe why these discontinuities exist.

# 9

A.APR.D.6

F.IF.C.7.D

Identify features of rational functions with equal degrees in the numerator and the denominator. Describe how to calculate features of these types of rational functions algebraically.

# 10

A.APR.D.6

F.IF.C.7.D

Identify features of rational functions with a larger degree in the denominator than in the numerator. Describe how to calculate these features algebraically.

# 11

A.APR.D.6

F.IF.C.7.D

Identify features of rational functions with a larger degree in the numerator than in the denominator. Describe how to calculate these features algebraically.

# 12

A.APR.D.6

F.IF.C.7.D

Analyze the graph and equations of rational functions and identify features. Use features of a rational function to identify and construct appropriate equations and graphs.

# 13

F.BF.B.3

F.IF.C.7.D

Describe transformations of rational functions.

A.REI.A.2

# 15

A.REI.A.2

Solve radical equations and identify extraneous solutions.

# 16

A.APR.D.6

A.REI.A.2

A.REI.D.11

Solve rational equations.

# 17

A.APR.D.6

A.CED.A.2

N.Q.A.1

Write and solve rational functions for contextual situations.

# 18

A.APR.D.6

A.CED.A.2

A.REI.A.2

Analyze rational and radical functions in context and write rational functions for percent applications.

## Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

### Core Standards

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##### 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.

• 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.

• 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.

##### 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.

##### 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.

• 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.

• 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.

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• A.APR.A.1

• F.BF.B.3

• F.BF.B.4.A

• A.CED.A.4

• 8.EE.A.1

• F.IF.A.1

• F.IF.B.4

• F.IF.C.8

• F.IF.C.8.A

• A.REI.A.1

• A.SSE.A.1

### 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.