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Module 3: Rational and exponential functions

Topic A: Lesson 1: Solutions to polynomial equations

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Factoring higher-degree polynomials
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Factoring sum of cubes
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Factoring difference of cubes
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Complex numbers & sum of squares factorization
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Factoring sum of squares
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Finding zeros of polynomials (1 of 2)
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Finding zeros of polynomials (2 of 2)
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Finding zeros of polynomials (example 2)
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Practice
Factor polynomials: complex numbersGet 3 of 4 questions to level up!
Zeros of polynomials (with factoring)Get 3 of 4 questions to level up!

Topic A: Lessons 2-3: Roots of complex numbers

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Powers of complex numbers
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Complex number equations: x³=1
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Visualizing complex number powers
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Practice
Powers of complex numbersGet 3 of 4 questions to level up!

Topic A: Lessons 4-5: The binomial theorem

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Intro to the Binomial Theorem
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Pascal's triangle and binomial expansion
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Expanding binomials
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Expanding binomials w/o Pascal's triangle
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Binomial expansion & combinatorics
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Pascal's triangle & combinatorics
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Practice
Expand binomialsGet 3 of 4 questions to level up!

Quiz 1

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Topic A: Lessons 6-7: Ellipses

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Intro to ellipses
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Ellipse standard equation from graph
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Ellipse graph from standard equation
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Foci of an ellipse from equation
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Ellipse features review
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Ellipse equation review
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Ellipse foci review
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Practice
Graph & features of ellipsesGet 3 of 4 questions to level up!
Center & radii of ellipses from equationGet 3 of 4 questions to level up!
Ellipse standard equation & graphGet 3 of 4 questions to level up!
Foci of an ellipse from radiiGet 3 of 4 questions to level up!
Foci of an ellipse from equationGet 3 of 4 questions to level up!
Equation of an ellipse from featuresGet 3 of 4 questions to level up!

Quiz 2

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Topic A: Lesson 8: Hyperbolas

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Intro to hyperbolas
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Vertices & direction of a hyperbola
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Vertices & direction of a hyperbola (example 2)
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Foci of a hyperbola from equation
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Proof of the hyperbola foci formula
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Equation of a hyperbola not centered at the origin
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Practice
Vertices & direction of a hyperbolaGet 3 of 4 questions to level up!
Foci of a hyperbola from equationGet 3 of 4 questions to level up!
Equation of a hyperbola from featuresGet 3 of 4 questions to level up!

Quiz 3

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Topic A: Lesson 9: Volume and Cavalieri’s principle

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Lesson 9: No content yet
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Topic B: Lessons 10-11: Simplifying rational expressions

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Intro to rational expression simplification
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Intro to rational expressions
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Intro to simplifying rational expressions
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Simplifying rational expressions: common monomial factors
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Simplifying rational expressions: common binomial factors
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Simplifying rational expressions: opposite common binomial factors
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Simplifying rational expressions (advanced)
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Simplifying rational expressions: grouping
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Simplifying rational expressions: higher degree terms
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Simplifying rational expressions: two variables
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Practice
Simplify rational expressions: common monomial factorsGet 3 of 4 questions to level up!
Simplify rational expressions: common binomial factorsGet 3 of 4 questions to level up!
Simplify rational expressions (advanced)Get 3 of 4 questions to level up!

Quiz 4

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Topic B: Lessons 10-11: Multiplying & dividing rational expressions

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Multiplying & dividing rational expressions: monomials
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Multiplying rational expressions
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Dividing rational expressions
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Multiplying rational expressions
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Dividing rational expressions
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Multiplying rational expressions: multiple variables
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Dividing rational expressions: unknown expression
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Practice
Multiply & divide rational expressions (basic)Get 3 of 4 questions to level up!
Multiply & divide rational expressionsGet 3 of 4 questions to level up!
Multiply & divide rational expressions (advanced)Get 3 of 4 questions to level up!

Quiz 5

Level up on the above skills and collect up to 300 Mastery points

Topic B: Lessons 10-11: Adding & subtracting rational expressions

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Adding & subtracting rational expressions: like denominators
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Intro to adding & subtracting rational expressions
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Intro to adding rational expressions with unlike denominators
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Adding rational expression: unlike denominators
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Subtracting rational expressions: unlike denominators
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Least common multiple
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Least common multiple: repeating factors
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Subtracting rational expressions: factored denominators
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Least common multiple of polynomials
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Adding & subtracting rational expressions (advanced)
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Subtracting rational expressions
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Nested fractions
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Practice
Add & subtract rational expressions: like denominatorsGet 3 of 4 questions to level up!
Add & subtract rational expressions: unlike denominatorsGet 3 of 4 questions to level up!
Least common multipleGet 3 of 4 questions to level up!
Add & subtract rational expressions: factored denominatorsGet 3 of 4 questions to level up!
Add & subtract rational expressionsGet 3 of 4 questions to level up!
Nested fractionsGet 3 of 4 questions to level up!

Quiz 6

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Topic B: Lessons 12-13: End behavior and asymptotes of rational functions

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End behavior of rational functions
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Discontinuities of rational functions
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Analyzing vertical asymptotes of rational functions
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Practice
End behavior of rational functionsGet 3 of 4 questions to level up!
Rational function points of discontinuityGet 3 of 4 questions to level up!
Analyze vertical asymptotes of rational functionsGet 3 of 4 questions to level up!

Topic B: Lessons 14-15: Graphing rational functions

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Graphing rational functions according to asymptotes
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Graphs of rational functions: y-intercept
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Graphs of rational functions: horizontal asymptote
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Graphs of rational functions: vertical asymptotes
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Graphs of rational functions: zeros
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Graphing rational functions 1
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Graphing rational functions 2
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Graphing rational functions 3
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Graphing rational functions 4
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Lesson 15: No content yet
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Practice
Graphs of rational functionsGet 3 of 4 questions to level up!

Quiz 7

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Topic B: Lessons 16-17: Function composition

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Intro to composing functions
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Intro to composing functions
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Composing functions
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Evaluating composite functions
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Evaluating composite functions: using tables
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Evaluating composite functions: using graphs
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Finding composite functions
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Evaluating composite functions (advanced)
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Modeling with composite functions
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Modeling with composite functions (example 2)
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Practice
Evaluate composite functionsGet 3 of 4 questions to level up!
Evaluate composite functions: graphs & tablesGet 3 of 4 questions to level up!
Find composite functionsGet 3 of 4 questions to level up!
Model with composite functionsGet 3 of 4 questions to level up!

Quiz 8

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Topic C: Lesson 18: Inverse functions

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Intro to inverse functions
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Intro to inverse functions
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Inputs & outputs of inverse functions
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Graphing the inverse of a linear function
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Finding inverse functions: linear
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Finding inverse functions: quadratic
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Finding inverse functions: quadratic (example 2)
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Finding inverse functions: radical
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Finding inverses of rational functions
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Finding inverse functions
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Practice
Evaluate inverse functionsGet 3 of 4 questions to level up!
Finding inverses of linear functionsGet 3 of 4 questions to level up!

Quiz 9

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Topic C: Lesson 19: Restricting the domain

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Verifying inverse functions by composition
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Verifying inverse functions by composition
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Verifying inverse functions by composition: not inverse
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Determining if a function is invertible
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Intro to invertible functions
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Restricting domains of functions to make them invertible
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Practice
Verify inverse functionsGet 3 of 4 questions to level up!
Determine if a function is invertibleGet 3 of 4 questions to level up!
Restrict domains of functions to make them invertibleGet 3 of 4 questions to level up!

Topic C: Lessons 20-21: Inverse relationship of exponentials and logarithms

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Relationship between exponentials & logarithms
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Relationship between exponentials & logarithms: graphs
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Relationship between exponentials & logarithms: tables
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Lesson 21: No content yet
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Practice
Relationship between exponentials & logarithmsGet 3 of 4 questions to level up!

Quiz 10

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Unit test

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About this unit

"Students revisit the fundamental theorem of algebra as they explore complex roots of polynomial functions. They use polynomial identities, the binomial theorem, and Pascal’s Triangle to find roots of polynomials and roots of unity. Students compare and create different representations of functions while studying function composition, graphing functions, and finding inverse functions." Eureka Math/EngageNY (c) 2015 GreatMinds.org