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.
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)
This assessment accompanies Unit 4 and should be given on the suggested assessment day or after completing the unit.
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Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
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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 |
F.IF.B.5
A.APR.D.6
Define rational functions. Identify domain restrictions of rational functions.
F.IF.C.7.B
F.BF.B.4
Identify domain restrictions algebraically for non-invertible functions.
F.IF.C.7.B
F.BF.B.3
Graph and transform square root and cubic root functions.
N.RN.A.2
F.IF.B.5
Write rational functions in equivalent radical form and identify domain restrictions of rational and radical functions.
N.RN.A.2
Write radical and rational exponent expressions in equivalent forms.
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
Add and subtract rational expressions.
F.IF.C.7.D
A.APR.D.6
Identify asymptotic discontinuities (also known as infinite discontinuities) and removable discontinuities in a rational function and describe why these discontinuities exist.
F.IF.C.7.D
A.APR.D.6
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.
F.IF.C.7.D
A.APR.D.6
Identify features of rational functions with a larger degree in the denominator than in the numerator. Describe how to calculate these features algebraically.
F.IF.C.7.D
A.APR.D.6
Identify features of rational functions with a larger degree in the numerator than in the denominator. Describe how to calculate these features algebraically.
F.IF.C.7.D
A.APR.D.6
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.
F.IF.C.7.D
F.BF.B.3
Describe transformations of rational functions.
A.REI.A.2
Solve simple radical equations.
A.REI.A.2
Solve radical equations and identify extraneous solutions.
A.APR.D.6
A.REI.A.2
A.REI.D.11
Solve rational equations.
N.Q.A.1
A.APR.D.6
A.CED.A.2
Write and solve rational functions for contextual situations.
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.
Key: Major Cluster Supporting Cluster Additional Cluster
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