Functions 11 - CEMC

GRADE 11

Functions 11

ONTARIO 2008

The tables below list the correspondence between the overall expectations of the Ontario Functions 11 (MCR3U) curriculum and the CEMC Grade 9/10/11 courseware.

Each section of each table is labelled with a dark heading containing a MCR3U overall expectation. The left-hand entries in a section are corresponding CEMC Grade 9/10/11 courseware strands and units. The right-hand side entries are all relevant courseware lessons within this courseware strand and unit.

The CEMC Grade 9/10/11 courseware has been designed with curricula from across Canada in mind. It is not an exact match to the current curriculum in any specific jurisdiction. In order to help teachers and students determine any discrepancies relevant to them, the table below also includes all of the courseware lesson goals for any cited courseware lesson. Additionally, some italicized notes point out topics that are not covered by the courseware or covered in an earlier or later part of the CEMC courseware suite.

Characteristics of Functions: Representing Functions

Lesson 1: Introduction to Functions ? Represent relations in a variety of ways, including mapping diagrams, equations, sets

of ordered pairs, and graphs. ? Represent relations whose graphs are circles, by using equations, tables, and graphs. ? Identify when a relation is a function, by using the definition of a function or the

Vertical Line Test.

Introduction to Functions

Unit 1: Representing

Functions

Lesson 2: Function Notation ? Describe functions using function notation. ? Analyze linear functions using function notation. ? Analyze quadratic functions using function notation.

Lesson 3: Domain and Range ? Describe functions using function notation. ? Analyze linear functions using function notation. ? Analyze quadratic functions using function notation.

Lesson 4: Domain and Range of Two New Functions ? Describe functions using function notation. ? Analyze linear functions using function notation. ? Analyze quadratic functions using function notation.

Introduction to Functions

Unit 2: Transforming and Graphing Functions

Lesson 1: Graphing Three Common Functions

?

Sketch

the

graphs

of

()

=

2,

()

=

,

and

()

=

1.

? Introduce the idea of an asymptote on a graph.

?

Identify

the

domain

and

range

of

the

functions

()

=

2,

()

=

,

and

()

=

1

using their graphs.

Lesson 2: Functions and Translations ? Define horizontal and vertical translations, and explore the effects of these

transformations on graphs. ? Observe the effect of horizontal and vertical translations on the domain and range of a

function. ? Express horizontal and vertical translations in function notation.

? Sketch the graph of a function by applying horizontal and vertical translations to a

base graph.

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Lesson 3: Horizontal Stretches, Compressions, and Reflections ? Describe how a reflection in the y-axis affects a function, and express this type of

transformation in function notation. ? Describe how a horizontal stretch or compression affects a function, and express this

type of transformation in function notation. ? Sketch graphs by applying a reflection in the y-axis, and/or a horizontal stretch or

compression to a known graph of a function.

? Identify the domain and range of a function, after a horizontal stretch or compression

and/or reflection in the y-axis.

Lesson 4: Vertical Stretches, Compressions, and Reflections ? Describe how a reflection in the x-axis affects a function, and express this type of

reflection in function notation. ? Describe how a vertical stretch or compression affects a function, and express this

type of transformation in function notation. ? Sketch graphs by applying a reflection in the x-axis, and/or a vertical stretch or

compression to a known graph of a function. ? Identify the domain and range of a function after a vertical stretch or compression

and/or reflection in the x-axis.

? Compare reflections in the x-axis with reflections in the y-axis, and compare vertical

stretches/compressions to horizontal stretches/compressions.

Lesson 5: Combining Transformations ? Identify the transformations that are applied to the graph of y=f(x) to obtain the graph

of y=af(b(x-h))+k. ? Sketch the graph of a function by applying transformations to a base graph in an

appropriate order.

? Identify the domain and range of a transformed function.

Introduction to Functions

Unit 3: Inverses of Functions

Lesson 1: Introduction to Inverses ? Determine the inverse of a function given tables or mapping diagrams. ? Determine the relationship between the graph of a function and the graph of

its inverse.

? Determine values of the inverse of f(x) given an algebraic expression for f(x).

Lesson 2: Determining Inverses of Linear Functions Algebraically ? Determine the inverse of a linear function algebraically.

? Determine the domain and range of the inverse of a function.

Lesson 3: Inverses of Quadratic Functions ? Determine if the inverse of a function is a function. ? Calculate the inverse of a quadratic function algebraically.

? Restrict the domain of a quadratic function so that the inverse is a function.

Characteristics of Functions: Solving Problems Involving Quadratic Functions

Introduction to Functions

Unit 1: Representing

Functions

Lesson 2: Function Notation ? Describe functions using function notation. ? Analyze linear functions using function notation. ? Analyze quadratic functions using function notation.

Quadratic Relations

Unit 5: Solving Problems Involving Quadratic

Relations

Lesson 3: The Number of Zeros of a Quadratic Relation ? Determine the number of zeros of a quadratic relation given its equation written in

factored or vertex form. ? Calculate the discriminant of a quadratic relation given in standard form and use it to

determine the number of zeros of the relation. ? Given a family of parabolas, determine which members of the family have 0, 1, or

2 zeros.

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Note: Review of quadratic concepts can be found in earlier Quadratic Relations units.

Lesson 4: Intersections of Linear and Quadratic Relations ? Identify the possible number of points of intersection between a linear relation and a

quadratic relation. ? Identify the point(s) of intersection between a linear relation and a quadratic relation

both graphically and algebraically. ? Use the discriminant to determine the number of point(s) of intersection between a

linear relation and a quadratic relation.

Lesson 5: Applications ? Use partial factoring to determine the vertex of a quadratic relation. ? Solve problems involving substitution into a quadratic relation. ? Solve problems that require solving a quadratic equation. ? Solve problems that involve finding the maximum or minimum of a quadratic relation. ? Select an appropriate computational strategy depending on the problem.

Characteristics of Functions: Determining Equivalent Algebraic Expressions

Number Sense and Algebraic

Expressions

Unit 2: Manipulating

Algebraic Expressions

Lesson 4: Multiplying a Polynomial by a Polynomial ? Apply the distributive property to multiply a polynomial by a polynomial.

Lesson 5: Simplifying Polynomials ? Simplify polynomials by adding, subtracting, and multiplying. ? Define the term equivalence. ? Determine if two algebraic expressions are equivalent.

Lesson 1: Introduction to Radicals ? Simplify and order radicals involving integers and rational numbers. ? Use technology to estimate the value of a radical. ? Recognize the difference between exact and approximate values.

Number Sense and Algebraic

Expressions

Unit 3: Radicals and Rational Functions

Lesson 2: Operations With Radicals ? Add, subtract, and multiply to simplify radical expressions. ? Simplify radical expressions by rationalizing the denominator.

Lesson 4: Introduction to Rational Expressions ? Define rational expressions. ? State restrictions on the variable values in a rational expression. ? Simplify rational expressions. ? Determine equivalence in rational expressions.

Lesson 5: Multiplying and Dividing Rational Expressions ? Multiply and divide rational expressions. ? Simplify these expressions and state restrictions on the variable values.

Lesson 6: Adding and Subtracting Rational Expressions ? Determine a common denominator for rational expressions. ? Add and subtract rational expressions and state restrictions on the variable(s). ? Simplify rational expressions involving various operations.

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Exponential Functions: Representing Exponential Functions

Number Sense and Algebraic Expressions

Unit 1: Exponents

Lesson 5: Rational Exponents -- Part 1

? Define the principal nth root of a number.

?

Explore rational exponents of the form 1.

Lesson 6: Rational Exponents -- Part 2 ? Simplify and evaluate positive rational exponents of the form . ? Simplify and evaluate negative rational exponents of the form-.

Lesson 7: Exponent Laws All Together ? Simplify algebraic expressions. ? Evaluate numerical expressions.

Exponential and Trigonometric Functions

Unit 1: Exponential Functions

Lesson 1: Introduction to Exponential Functions ? Define exponential growth and exponential decay and determine a function

describing these processes. ? Evaluate an exponential function with a particular input to determine the outcome

of an exponential growth or decay process.

Lesson 2: Properties of Basic Exponential Functions ? Determine if an exponential function of the form f(x)=acx can be evaluated when x

is o a negative integer, or o a rational number.

? Determine the possible values that an exponential function of the form f(x)=acx with a>0 can take.

? Locate any intercepts and asymptotes of an exponential function of the form f(x)=acx with a>0.

Exponential Functions: Connecting Graphs and Equations of Exponential Functions

Lesson 3: Identifying Exponential Functions ? Determine the equation of an exponential function given

o a table of values, or o a graph. ? Use finite differences to determine if a data set is representative of an exponential process.

Exponential and Trigonometric Functions

Unit 1: Exponential Functions

Lesson 4: Transformations of Exponential Functions ? Determine an equation for an exponential function that has undergone reflection,

stretch, and translation transformations. ? Determine the domain and range of a transformed exponential function. ? Graph a transformed exponential curve.

Lesson 5: Comparing Exponential Functions ? Express a given exponential function in a different base. ? Find an exponential function with a base greater than one which describes

exponential decay.

Lesson 6: Modelling With Exponential Functions ? Given a description of an exponential process, determine an appropriate form for

a transformed exponential function which models this process. ? Fix the base and parameters of a transformed exponential function so as

to accurately describe an exponential process.

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Exponential Functions: Solving Problems Involving Exponential Functions

Exponential and Trigonometric Functions

Unit 1: Exponential Functions

Lesson 3: Identifying Exponential Functions ? Determine the equation of an exponential function given

o a table of values, or o a graph. ? Use finite differences to determine if a data set is representative of an exponential process.

Lesson 5: Comparing Exponential Functions ? Express a given exponential function in a different base. ? Find an exponential function with a base greater than one which describes

exponential decay.

Lesson 6: Modelling With Exponential Functions ? Given a description of an exponential process, determine an appropriate form for

a transformed exponential function which models this process. ? Fix the base and parameters of a transformed exponential function so as

to accurately describe an exponential process.

Discrete Functions: Representing Sequences

Sequences, Series, and Financial Literacy

Lesson 1: Introducing Sequences ? Express sequences numerically and graphically, using term notation. ? Represent sequences algebraically, using a recursion formula. ? Represent sequences algebraically, using a general term or function notation. ? Make connections between the different algebraic representations of sequences.

Unit 1: Representing Sequences

Lesson 2: Pascal's Triangle and Binomial Expansions ? Generate Pascal's triangle. ? Identify patterns in Pascal's triangle. ? Expand powers of binomials, (a+b)n. ? Determine specific terms in the expansion of (a+b)n.

Discrete Functions: Investigating Arithmetic and Geometric Sequences and Series

Lesson 1: Arithmetic Sequences ? Identify if a sequence is arithmetic. ? Determine a recursive formula for an arithmetic sequence. ? Determine the general term of an arithmetic sequence. ? Solve questions about arithmetic sequences using the general term.

Sequences, Series, and Financial Literacy

Unit 2: Arithmetic and Geometric

Sequences and Series and Financial Applications

Lesson 3: Geometric Sequences ? Identify a geometric sequence. ? Determine a recursion formula for a geometric sequence. ? Determine and apply the formula for the general term of a geometric sequence. ? Solve problems involving geometric sequences.

Lesson 5: Arithmetic Series ? Define a series as the sum of the terms of a sequence. ? Derive two formulas for the sum of the first n terms of an arithmetic series. ? Solve problems using the formulas for the sum of the first n terms of an arithmetic

series.

Lesson 6: Geometric Series ? Define a geometric series. ? Derive a formula for the sum of the first n terms in a geometric series (Sn) and use

this formula to calculate sums of given geometric series. ? Solve problems involving the application of geometric series.

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