Ontario · Grade 11
Functions, Grade 11 (University)
- MCR3U
Ontario’s Grade 11 math course, from Characteristics of Functions to Trigonometric Functions.
- Units
- 4
- Lessons
- 13
- Skills
- 68
- Provincial exam
- None
- Official curriculum
- Ontario Ministry of Education
Unit by unit
Every skill in the course, numbered by unit.
Unit 1 · 18 skills
Characteristics of Functions
Representing Functions
What makes a relation a function
I can explain what a function is and tell it apart from a relation that isn't one.
Skill 1.1
Using function notation and evaluating functions
I can use function notation and evaluate functions.
Skill 1.2
Explaining domain and range
I can explain domain and range and describe real-world restrictions on them.
Skill 1.3
Connecting a function's inverse to reverse processes
I can connect finding a function's inverse to reversing a process.
Skill 1.4
Finding a function's inverse numerically and graphically
I can find a function's inverse numerically and graphically, and connect their graphs.
Skill 1.5
Domain and range of an inverse relation
I can relate the domain and range of a function to those of its inverse, and check if the inverse is a function.
Skill 1.6
Finding a function's inverse algebraically
I can find a function's inverse algebraically.
Skill 1.7
Transformations of Functions
The roles of a, k, d, and c in y = af(k(x − d)) + c
I can describe the roles of a, k, d, and c in y = af(k(x − d)) + c.
Skill 1.8
Sketching transformed functions
I can sketch transformed functions and state their domain and range.
Skill 1.9
Solving Problems Involving Quadratic Functions
Finding the number of zeros of a quadratic function
I can find the number of zeros of a quadratic function.
Skill 1.10
Finding the maximum or minimum of a quadratic function algebraically
I can find the maximum or minimum value of a quadratic function algebraically.
Skill 1.11
Solving real-world quadratic function problems
I can solve real-world problems involving quadratic functions.
Skill 1.12
The family of quadratic functions sharing the same zeros
I can investigate the family of quadratic functions with the same zeros and find one given a point.
Skill 1.13
Solving the intersection of a linear and a quadratic function
I can solve for the intersection of a linear and a quadratic function.
Skill 1.14
Determining Equivalent Algebraic Expressions
Simplifying polynomial expressions
I can simplify polynomial expressions by adding, subtracting, and multiplying.
Skill 1.15
Simplifying radicals and radical expressions
I can simplify radicals and radical expressions.
Skill 1.16
Simplifying rational expressions and stating restrictions
I can simplify rational expressions and state their restrictions.
Skill 1.17
Checking whether two algebraic expressions are equivalent
I can check whether two algebraic expressions are equivalent.
Skill 1.18
The official wording · 18 outcomes
- 1.1
- explain the meaning of the term function, and distinguish a function from a relation that is not a function, through investigation of linear and quadratic relations using a variety of repre- sentations
- 1.2
- represent linear and quadratic functions using function notation, given their equations, tables of values, or graphs, and substitute into and evaluate functions
- 1.3
- explain the meanings of the terms domain and range, through investigation using numer- ic, graphical, and algebraic representations
- 1.4
- relate the process of determining the inverse of a function to their understanding of reverse processes
- 1.5
- determine, through investigation, the numeric or graphical represen- tation of the inverse of a linear or quadratic function, given the numeric, graphical, or algebraic representation of the function, and make connections
- 1.6
- determine, through investigation, the relation- ship between the domain and range of a func- tion and the domain and range of the inverse relation, and determine whether or not the inverse relation is a function
- 1.7
- determine, using function notation when appropriate, the algebraic representation of the inverse of a linear or quadratic function, given the algebraic representation of the function
- 1.8
- determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y = af(k(x – d)) + c, and describe these roles in terms of transformations on the graphs
- 1.9
- sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graphs of f(x) = x
- 1.10
- determine the number of zeros (i.e., x-intercepts) of a quadratic function, using a variety of strategies (e.g., inspecting graphs; factoring; calculating the discriminant)
- 1.11
- determine the maximum or minimum value of a quadratic function whose equation is given in the form f(x) = ax + bx + c, using an algebraic method
- 1.12
- solve problems involving quadratic functions arising from real-world applications and represented using function notation
- 1.13
- determine, through investigation, the trans- formational relationship among the family of quadratic functions that have the same zeros, and determine the algebraic representation of a quadratic function, given the real roots of the corresponding quadratic equation and a point on the function
- 1.14
- solve problems involving the intersection of a linear function and a quadratic function graphically and algebraically
- 1.15
- simplify polynomial expressions by adding, subtracting, and multiplying
- 1.16
- verify, through investigation with and without technology, that √ab = √a x √b, a ≥ 0, b ≥ 0, and use this relationship to simplify radicals
- 1.17
- simplify rational expressions by adding, subtracting, multiplying, and dividing, and state the restrictions on the variable values
- 1.18
- determine if two given algebraic expressions are equivalent (i.e., by simplifying; by substituting values)
Key ideas
A function assigns exactly one output to each input — no input can have two different outputs.
a and k control stretches/compressions and reflections (a vertically, k horizontally).
A quadratic function can have 0, 1, or 2 real zeros (x-intercepts).
Adding and subtracting polynomials means combining like terms.
The vertical-line test on a graph, or checking for a one-to-one or many-to-one mapping, both reveal whether a relation is a function.
Watch out for
- Thinking a relation with repeated y-values (many-to-one) can't be a function, when it actually can.
- Expecting d to shift the graph in the same direction as its sign inside the brackets.
- Assuming a quadratic always has exactly two real zeros.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 2 · 12 skills
Exponential Functions
Representing Exponential Functions
Graphing an exponential relation and defining it as a function
I can graph an exponential relation and define it as a function.
Skill 2.1
Finding the value of a power with a rational exponent
I can find the value of a power with a rational exponent.
Skill 2.2
Simplifying and evaluating expressions with integer and rational exponents
I can simplify and evaluate expressions with integer and rational exponents.
Skill 2.3
Connecting Graphs and Equations of Exponential Functions
Key properties of exponential functions
I can describe the key properties of exponential functions.
Skill 2.4
Distinguishing exponential from linear and quadratic functions
I can distinguish exponential functions from linear and quadratic functions.
Skill 2.5
The roles of a, k, d, and c in exponential transformations
I can describe the roles of a, k, d, and c in exponential transformations.
Skill 2.6
Sketching transformed exponential functions
I can sketch transformed exponential functions and state their domain and range.
Skill 2.7
Expressing an exponential function using a different base
I can express an exponential function's equation using a different base.
Skill 2.8
Writing an exponential function's equation from its graph or properties
I can write an exponential function's equation from its graph or properties.
Skill 2.9
Solving Problems Involving Exponential Functions
Collecting and graphing exponential data
I can collect data that forms an exponential function and graph it.
Skill 2.10
Identifying real-world exponential growth and decay
I can identify real-world exponential growth and decay situations.
Skill 2.11
Solving real-world exponential function problems
I can solve real-world exponential function problems.
Skill 2.12
The official wording · 12 outcomes
- 2.1
- graph, with and without technology, an expo- nential relation, given its equation in the form y = a (a > 0, a ≠ 1), define this relation as the function f(x) = a , and explain why it is a function
- 2.2
- determine, through investigation using a variety of tools (e.g., calculator, paper and pencil, graphing technology) and strategies (e.g., patterning; finding values from a graph; interpreting the exponent laws), the value of a power with a rational exponent
- 2.3
- simplify algebraic expressions containing integer and rational exponents
- 2.4
- determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes
- 2.5
- distinguish exponential functions from linear and quadratic functions by making compar- isons in a variety of ways
- 2.6
- determine, through investigation using tech- nology, the roles of the parameters a, k, d, and c in functions of the form y = af(k(x – d)) + c, and describe these roles in terms of transfor- mations on the graph of f(x) = a (a > 0, a ≠ 1)
- 2.7
- sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graph of f(x) = a (a > 0, a ≠ 1), and state the domain and range of the transformed functions
- 2.8
- determine, through investigation using techno- logy, that the equation of a given exponential function can be expressed using different bases
- 2.9
- represent an exponential function with an equation, given its graph or its properties
- 2.10
- collect data that can be modelled as an expo- nential function, through investigation with and without technology, from primary sources
- 2.11
- identify exponential functions, including those that arise from real-world applications involving growth and decay
- 2.12
- solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications
Key ideas
An exponential relation y = aˣ (a > 0, a ≠ 1) graphs as a curve that grows or decays rapidly.
Exponential functions have domain = all real numbers and range = positive real numbers only (for the basic form).
Real experiments (like cooling water) or secondary data sources can produce exponential-shaped data.
The relation passes the vertical-line test, so it can be defined as the function f(x) = aˣ.
Exponential functions are always increasing or always decreasing throughout their domain, and they have a horizontal asymptote.
Watch out for
- Confusing an exponential function's rapid growth with a quadratic function's growth.
- Thinking the range includes zero or negative numbers, when the basic exponential function's range is only positive reals.
- Assuming any decreasing curve must be exponential decay without checking.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 3 · 18 skills
Discrete Functions
Representing Sequences
Connecting sequences to discrete functions
I can connect sequences to discrete functions and tell discrete from continuous.
Skill 3.1
Describing a recursive procedure for a sequence
I can describe a recursive procedure for generating a sequence.
Skill 3.2
Connecting the nth-term formula to function notation
I can connect the nth-term formula of a sequence to function notation.
Skill 3.3
Representing a sequence algebraically in multiple ways
I can represent a sequence algebraically in multiple ways and explain what each shows.
Skill 3.4
Recursive patterns in the Fibonacci sequence and Pascal's triangle
I can investigate recursive patterns in the Fibonacci sequence and Pascal's triangle.
Skill 3.5
Pascal's triangle and expanding binomials
I can use Pascal's triangle to expand binomials.
Skill 3.6
Investigating Arithmetic and Geometric Sequences and Series
Identifying a sequence as arithmetic, geometric, or neither
I can identify a sequence as arithmetic, geometric, or neither.
Skill 3.7
Finding the general-term formula for arithmetic and geometric sequences
I can find the general-term formula for arithmetic and geometric sequences.
Skill 3.8
Finding the sum formula for arithmetic and geometric series
I can find the sum formula for arithmetic and geometric series.
Skill 3.9
Solving problems with arithmetic and geometric sequences and series
I can solve problems involving arithmetic and geometric sequences and series.
Skill 3.10
Solving Problems Involving Financial Applications
Connecting simple interest to arithmetic sequences and linear growth
I can connect simple interest to arithmetic sequences and linear growth.
Skill 3.11
Connecting compound interest to geometric sequences and exponential growth
I can connect compound interest to geometric sequences and exponential growth.
Skill 3.12
Solving for amount, principal, or interest rate with the compound interest formula
I can solve for amount, principal, or interest rate using the compound interest formula.
Skill 3.13
Solving for the number of compounding periods
I can solve for the number of compounding periods using the compound interest formula.
Skill 3.14
What an annuity is, with real examples
I can explain what an annuity is, with real examples.
Skill 3.15
Connecting annuities to geometric series and exponential growth
I can connect annuities to geometric series and exponential growth.
Skill 3.16
Effects of changing an annuity's conditions
I can investigate the effects of changing an annuity's conditions.
Skill 3.17
Solving for the amount, present value, or payment of an annuity
I can solve for the amount, present value, or payment of an annuity.
Skill 3.18
The official wording · 18 outcomes
- 3.1
- make connections between sequences and discrete functions, represent sequences using function notation, and distinguish between a discrete function and a continuous function
- 3.2
- determine and describe (e.g., in words; using flow charts) a recursive procedure for gen- erating a sequence, given the initial terms
- 3.3
- connect the formula for the nth term of a sequence to the representation in function notation, and write terms of a sequence given one of these representations or a recursion formula
- 3.4
- represent a sequence algebraically using a recursion formula, function notation, or the formula for the nth term
- 3.5
- determine, through investigation, recursive patterns in the Fibonacci sequence, in related sequences, and in Pascal’s triangle, and represent the patterns in a variety of ways
- 3.6
- determine, through investigation, and describe the relationship between Pascal’s triangle and the expansion of binomials, and apply the relationship to expand bino- mials raised to whole-number exponents
- 3.7
- identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representation
- 3.8
- determine the formula for the general term of an arithmetic sequence
- 3.9
- determine the formula for the sum of an arithmetic or geometric series, through inves- tigation using a variety of tools
- 3.10
- solve problems involving arithmetic and geo- metric sequences and series, including those arising from real-world applications
- 3.11
- make and describe connections between simple interest, arithmetic sequences, and linear growth, through investigation with technology
- 3.12
- make and describe connections between compound interest, geometric sequences, and exponential growth, through investiga- tion with technology
- 3.13
- solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV), the principal, P (also referred to as present value, PV), or the interest rate per compounding period, i, using the compound interest formula
- 3.14
- determine, through investigation using technology (e.g., scientific calculator, the TVM Solver on a graphing calculator, online tools), the number of compounding periods, n, using the compound interest formula
- 3.15
- explain the meaning of the term annuity, and determine the relationships between ordinary simple annuities (i.e., annuities in which pay- ments are made at the end of each period, and compounding and payment periods are the same)
- 3.16
- through investigation with techno- logy (e.g., use a spreadsheet to determine and graph the future value of an ordinary simple annuity for varying numbers of compounding periods; investigate how the contributions of each payment to the future value of an ordi- nary simple annuity are related to the terms of a geometric series)
- 3.17
- determine, through investigation using technology (e.g., the TVM Solver on a graph- ing calculator, online tools), the effects of changing the conditions (i.e., the payments, the frequency of the payments, the interest rate, the compounding period) of ordinary simple annuities
- 3.18
- solve problems, using technology (e.g., scien- tific calculator, spreadsheet, graphing calcula- tor), that involve the amount, the present value, and the regular payment of an ordinary simple annuity
Key ideas
A sequence is a discrete function — its domain is only the natural numbers (or a subset), so its graph is separate dots, not a connected line.
An arithmetic sequence has a constant DIFFERENCE between consecutive terms.
Simple interest adds the SAME dollar amount each period — exactly like an arithmetic sequence's constant difference.
The SAME formula can define a discrete function (domain = natural numbers) or a continuous function (domain = all real numbers), depending on context.
A geometric sequence has a constant RATIO between consecutive terms.
Watch out for
- Connecting the dots of a sequence's graph with a line, treating it as continuous.
- Checking only the first pair of terms instead of confirming the pattern holds throughout.
- Assuming any interest calculation must be compound, without checking if the growth is actually linear.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 4 · 20 skills
Trigonometric Functions
Determining and Applying Trigonometric Ratios
Exact trig values for the special angles
I can find the exact trig values for the special angles.
Skill 4.1
Finding trig values for angles from 0° to 360°
I can find trig values for angles from 0° to 360°.
Skill 4.2
Finding two angles with the same trig ratio value
I can find two angles between 0° and 360° with the same trig ratio value.
Skill 4.3
Defining the secant, cosecant, and cotangent ratios
I can define the secant, cosecant, and cotangent ratios.
Skill 4.4
Proving simple trigonometric identities
I can prove simple trigonometric identities using the Pythagorean identity.
Skill 4.5
Solving right and oblique triangle problems in two dimensions
I can solve right and oblique triangle problems in two dimensions.
Skill 4.6
Solving right and oblique triangle problems in three dimensions
I can solve right and oblique triangle problems in three dimensions.
Skill 4.7
Connecting Graphs and Equations of Sinusoidal Functions
Key properties of periodic functions
I can describe the key properties of periodic functions.
Skill 4.8
Predicting future behaviour of a periodic relationship
I can predict the future behaviour of a periodic relationship.
Skill 4.9
Connecting the sine and cosine ratios to the sine and cosine functions
I can connect the sine and cosine ratios to the sine and cosine functions.
Skill 4.10
Sketching sine and cosine graphs and their key properties
I can sketch sine and cosine graphs and describe their key properties.
Skill 4.11
The roles of a, k, d, and c in sinusoidal transformations
I can describe the roles of a, k, d, and c in sinusoidal transformations.
Skill 4.12
Finding amplitude, period, phase shift, domain, and range from an equation
I can find amplitude, period, phase shift, domain, and range from a sinusoidal equation.
Skill 4.13
Sketching transformed sine and cosine graphs
I can sketch transformed sine and cosine graphs.
Skill 4.14
Writing a sinusoidal function's equation from its graph or properties
I can write a sinusoidal function's equation from its graph or properties.
Skill 4.15
Solving Problems Involving Sinusoidal Functions
Collecting and graphing sinusoidal data
I can collect data that forms a sinusoidal function and graph it.
Skill 4.16
Identifying real-world periodic and sinusoidal functions
I can identify real-world periodic and sinusoidal functions.
Skill 4.17
Using sinusoidal functions to model phenomena without angles
I can use sinusoidal functions to model phenomena without angles, like time-based cycles.
Skill 4.18
Predicting the effect of varying conditions on a periodic model
I can predict the effects of varying conditions on a periodic model.
Skill 4.19
Solving real-world sinusoidal function problems
I can pose and solve real-world problems involving sinusoidal functions.
Skill 4.20
The official wording · 20 outcomes
- 4.1
- determine the exact values of the sine, cosine, and tangent of the special angles: 0º, 30º, 45º, 60º, and 90º
- 4.2
- determine the values of the sine, cosine, and tangent of angles from 0º to 360º, through investigation using a variety of tools
- 4.3
- determine the measures of two angles from 0º to 360º for which the value of a given trigonometric ratio is the same
- 4.4
- define the secant, cosecant, and cotangent ratios for angles in a right triangle in terms of the sides of the triangle
- 4.5
- prove simple trigonometric identities, using the Pythagorean identity sin x + cos x = 1
- 4.6
- pose problems involving right triangles and oblique triangles in two- dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law
- 4.7
- pose problems involving right triangles and oblique triangles in three-dimensional set- tings, and solve these and other such pro- blems using the primary trigonometric ratios, the cosine law, and the sine law
- 4.8
- describe key properties (e.g., cycle, amplitude, period) of periodic functions arising from real-world applications
- 4.9
- predict, by extrapolating, the future behaviour of a relationship modelled using a numeric or graphical representation of a periodic function
- 4.10
- make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios
- 4.11
- sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties
- 4.12
- determine, through investigation using tech- nology, the roles of the parameters a, k, d, and c in functions of the form y =af(k(x – d)) + c, where f(x) =sinx or f(x) =cosx with angles expressed in degrees, and describe these roles in terms of transformations
- 4.13
- determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form f(x) = asin(k(x – d)) + c
- 4.14
- sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graphs of f(x) =sinx and f(x) =cosx, and state the domain and range of the transformed functions
- 4.15
- represent a sinusoidal function with an equation, given its graph or its properties
- 4.16
- collect data that can be modelled as a sinu- soidal function (e.g., voltage in an AC circuit, sound waves), through investigation with and without technology
- 4.17
- identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena
- 4.18
- determine, through investigation, how sinu- soidal functions can be used to model periodic phenomena that do not involve angles
- 4.19
- predict the effects on a mathematical model (i.e., graph, equation) of an application involving periodic phenomena when the conditions in the application are varied
- 4.20
- pose problems based on applications involving a sinusoidal function, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation
Key ideas
0°, 30°, 45°, 60°, and 90° have EXACT sine, cosine, and tangent values (often involving √2 or √3), not just decimal approximations.
A periodic function repeats the same pattern (a cycle) over and over at regular intervals.
Real periodic phenomena (like a swinging pendulum or AC voltage) can be measured and graphed to reveal a sinusoidal shape.
These exact values come from specific right triangles (30-60-90 and 45-45-90) and are worth memorizing.
Amplitude measures how far the function swings from its middle value; period measures how long one cycle takes.
Watch out for
- Using a decimal approximation instead of the exact radical value when an exact value is required.
- Confusing amplitude (half the total swing) with the total range of the function.
- Assuming any wavy-looking data must be perfectly sinusoidal without checking amplitude/period consistency.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Plan the term
Pick your term and the days you study. Each unit gets its share of the weeks and comes back a week later for a review. The last two weeks go over the whole course.
Which course?
Ontario has 4 math courses for Grade 11. Here they are side by side.
- Foundations for College Mathematics, Grade 11 (College)MBF3C4 units · 57 skillsNo provincial examWhat it coversMathematical Models · Personal Finance · Geometry and Trigonometry · Data Management
- Functions and Applications, Grade 11 (University/College)MCF3M3 units · 53 skillsNo provincial examWhat it coversQuadratic Functions · Exponential Functions · Trigonometric Functions
- Functions, Grade 11 (University)This courseMCR3U4 units · 68 skillsNo provincial examWhat it coversCharacteristics of Functions · Exponential Functions · Discrete Functions · Trigonometric Functions
- Mathematics for Work and Everyday Life, Grade 11 (Workplace)MEL3E3 units · 48 skillsNo provincial examWhat it coversEarning and Purchasing · Saving, Investing, and Borrowing · Transportation and Travel
What a university, college or program asks for is theirs to say: check its admission page, and your school’s course guide.
The official curriculum
This page follows Ontario’s own curriculum for Functions, Grade 11 (University).
Textbooks
The textbooks Ontario Ministry of Education lists for Functions, Grade 11 (University):
- McGraw-Hill Ryerson Functions 11
- Nelson Functions 11
From the Trillium List, read October 27, 2026. ontario.ca
Strands
| Strand | Outcomes | Taught in |
|---|---|---|
| ACharacteristics of Functions | Outcomes 18 | Taught in Unit 1 |
| BExponential Functions | Outcomes 12 | Taught in Unit 2 |
| CDiscrete Functions | Outcomes 18 | Taught in Unit 3 |
| DTrigonometric Functions | Outcomes 20 | Taught in Unit 4 |
Study it in MapleMind
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Course Companion
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Questions about Functions, Grade 11 (University)
What is Functions, Grade 11 (University)?
Ontario’s Grade 11 math course: 4 units and 68 skills, from Characteristics of Functions to Trigonometric Functions.
What units are in Functions, Grade 11 (University)?
1. Characteristics of Functions, 2. Exponential Functions, 3. Discrete Functions and 4. Trigonometric Functions.
Is there a provincial exam for Functions, Grade 11 (University)?
No. Ontario gives no provincial exam for Functions, Grade 11 (University).
What textbook does Functions, Grade 11 (University) use?
The textbooks Ontario Ministry of Education lists for Functions, Grade 11 (University): McGraw-Hill Ryerson Functions 11, McGraw-Hill Ryerson and Nelson Functions 11, Nelson Education Ltd.
What comes after Functions, Grade 11 (University)?
Advanced Functions, Grade 12 (University), Calculus and Vectors, Grade 12 (University), Mathematics of Data Management, Grade 12 (University), Foundations for College Mathematics, Grade 12 (College), Mathematics for College Technology, Grade 12 (College) and Mathematics for Work and Everyday Life, Grade 12 (Workplace).
Can MapleMind help with Functions, Grade 11 (University)?
Yes. MapleMind teaches every skill of Functions, Grade 11 (University), step by step, with practice and checks, and its tutor answers questions about any of them. It’s free with daily limits, on iPhone, Android and the web.
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