ADVANCED FUNCTIONS MHF4U:
Essential Skills Review
ADVANCED FUNCTIONS MHF4U UNIT 1: Functions: Characteristics and Properties
1.1 Functions
Website Video Clips with Solutions:
Khan Academy Exercises:
1.2 Absolute Value
Website Video Clips with Solutions:
Khan Academy Exercises:
1.3 Properties of Graphs of
Functions
Website Video Clips with Solutions:
Pages 4 #
Introduction to Function Notations
Evaluating Functions
Evaluating Functions Example 1
Evaluating Functions Example 2
Function Notation
Go to topic “Algebra” and click on “Functions” then click on “Function Notation”
Review of Functions practice
Functions Practice 1
Functions Practice 2
Pages 14 #
Absolute Value examples
Absolute value explanation
Go to topic “Algebra” and click on “Functions” then click on “Function Notation”
Absolute Value example 1
Absolute Value example 2
Absolute Value practice basic
Absolute Value practice intermediate
Go to topic “Algebra” and click on “Positive and negative numbers” then click on “absolute value”
Pages 17 #
Properties of graphs of functions examples
Explanation of the seven types of graphs for elementary functions
1.4 Sketching Graphs of Functions
Website Video Clips with Solutions:
1.5 Inverse Relations
Website Video Clips with Solutions:
Pages 29 #
Sketching graphs of functions examples
Explanation of the seven types of graphs for elementary functions
Transforming the graphs of sine and cosine review
Graphing transformations of the function examples
Watch concept video and problems #1 to 3
Pages 38 #
Inverse relations examples
Finding the inverse examples
Watch concept video and problems #1 to 2
Proving that two functions are inverses
Watch concept video and problems #1 to 2
Finding an inverse graphically example
1.6 Piecewise Functions
Pages 46 #
Website Video Clips with Solutions:
Piecewise functions examples
Defined piecewise functions examples similar to question #10
1.7 Operations with Functions
Pages 54 #
Website Video Clips with Solutions:
Operations with functions examples
Unit 1 Review
Pages 60-62 #
ADVANCED FUNCTIONS MHF4U UNIT 2: Functions: Understanding Rates of Change
2.1 Average Rate of Change
Website Video Clips with Solutions:
2.2 Estimating Instantaneous Rates of
Change
Website Video Clips with Solutions:
2.3 Instantaneous Rates of Change
using Graphs
Website Video Clips with Solutions:
Additional Support tools:
Pages 76-78 #
Rates of Change and Slope of graph example
Average Rate of Change Examples
Watch Concept and Problem #1-3 video
Average Rate of Change connected to Graph
Application Problem height (distance) vs. time
Pages 86-88#
Instantaneous Rates of Change Examples
Watch Concept and Problem #1-3 video
Pages 91-92#
Exploring Rates of change using graphs
See Videos from 2.2 & 2.3
Graphing Software: Geogebra
Graphing software that you can use at home: select “Applet start”. From here, you can
enter equations and get the graph. You also have the choice to download the software for
free.
Graphing Calculator Emulator
Download the free graphing calculator software
2.4 Using Rates of Change to create a
Graphical Model
Website Video Clips with Solutions:
Pages 103-106#
Using rates of change to create graphical model examples
Rates of Change and Slope of graph example
Average Rate of Change Examples
Watch Concept and Problem #1-3 video
Instantaneous Rates of Change Examples
Watch Concept and Problem #1-3 video
2.5 Solving problems involving
Rates of Change
Pages 111-113
Website Video Clips with Solutions:
Solving problems involving rates of change examples
Review
Pages 10-11 #
ADVANCED FUNCTIONS MHF4U UNIT 3: Polynomial Functions
3.1 Exploring Polynomial Functions
Website Video Clips with Solutions:
Pages 127-128#
Polynomial Functions as Graphs (Power Graphs)
Review Polynomial skills from Grade 11 videos below
Additional Support tools:
Simplifying Polynomials Example 1
Simplifying Polynomials Example 2
Simplifying Application Polynomial example
Domain and Range
Adding and Subtracting Polynomial example
Go to topic “Algebra” and click on “Polynomials” then click on “Adding/Subtracting Polynomials”
Multiplying Polynomials Examples
Go to topic “Algebra” and click on “Polynomials” then click on “Multiplication with Polynomials”
Polynomial Factoring Examples
Go to topic “Algebra” and click on “Factoring” then click on “Grouping”
3.2 Characteristics of Polynomial
Functions
Website Video Clips with Solutions:
3.3 Characteristics of Factored
Polynomial Functions
Website Video Clips with Solutions:
3.4 Transformations of Cubic and
Quartic Functions
Pages 136-138 #
Characteristics of Polynomial Functions
Polynomial end behaviour
Watch concept video and problems 1 to 3
Polynomial Functions and their graphs
Pages 146-148 #
Graphing Polynomial Functions example
Watch concept video and problems 1 to 3
Finding the Equation of a polynomial from a graph
Watch concept video and problems 1 to 3
Finding Zeros of Polynomial Functions
Watch concept video and problems 1 to 3
Pages 155-158 #
Website Video Clips with Solutions:
Polynomial transformations graphical examples
Watch concept video and problems 1 to 3
Basic Transformation of Polynomial
3.5 Dividing Polynomials
Pages 168-170 #
Website Video Clips with Solutions:
Dividing Polynomials by long division
Watch concept video and problems 1 to 3
Dividing polynomials by synthetic division
Watch concept video and problems 1 to 2
Synthetic Division to evaluate Polynomials
Watch concept video and problems 1 to 2
Dividing by monomials and polynomials
Go to topic “Algebra” and click on “polynomials” then click on “Dividing by a monomial and dividing by
a polynomial”
3.6 Factoring Polynomials
Website Video Clips with Solutions:
3.7 Factoring a Sum and Difference
of Cubes
Website Video Clips with Solutions:
Pages 176-177 #
Factoring Review
Watch concept video and problems 1 to 3
Factoring Review 1
Watch concept video and problems 1 to 3
Factoring difficult functions examples
Watch concept video and problems 1 to 4
Pages 182#
Factoring sum and difference of cubes
Go to topic “Algebra” and click on “Factoring” then click on “Sum and difference”
Factoring sum and difference of cubes examples
Watch concept video and problems 1 to 3
Review
Pages 184-185 #
ADVANCED FUNCTIONS MHF4U UNIT 4: Polynomial Equations and Inequalities
4.1 Solving Polynomial Equations
Pages 204-206#
Solving with Synthetic Division Example
Watch concept video and problems 1
Website Video Clips with Solutions:
Factoring Review
Watch concept video and problems 1 to 3
Factoring Review 1
Watch concept video and problems 1 to 3
Factoring difficult functions examples
Watch concept video and problems 1 to 4
4.2 Solving Linear Inequalities
Pages 213-215 #
Website Video Clips with Solutions:
Linear Inequality
Watch concept video and problems 1 to 3
Double inequality examples
Inequality with one variable
Go to topic “Algebra” and click on “inequalities” then click on “linear, one variable”
Inequality with absolute value example
Go to topic “Algebra” and click on “inequalities” then click on “with absolute value”
Khan Academy Exercises:
Linear Inequality Practice
4.3 Solving Polynomial Inequalities
Pages 225-227 #
Website Video Clips with Solutions:
4.4 Rates of Change in Polynomial
Equations
Website Video Clips with Solutions:
Polynomial inequality examples
Go to topic “Algebra” and click on “inequalities” then click on “quadratic, with one variable”
Solving Polynomial using the sign chart example
Watch concept video and problems 1
Basic Graphing of Polynomials review
Watch concept video and problems 1
Pages 235-238 #
Rates of change in polynomial equations example
Review
Pages 240-241
ADVANCED FUNCTIONS MHF4U UNIT 5: Rational Functions, Equations and Inequalities
5.1 Graphs of Reciprocal Functions
Pages 424-425 # 2-5,8i,9,13,15-17
Website Video Clips with Solutions:
Graphs of reciprocal functions examples
Graphing reciprocal functions example 2
Watch Concept and Problem #1-3 videos
5.2 Exploring Quotients of
Polynomial function
Pages 430-431 #1-4,5i,6i,10-11 17-20
Website Video Clips with Solutions:
Polynomial quotient functions graphical examples
Watch Concept and Problem #1-3 videos
Quotients of polynomial functions
5.3 Graphs of Rational Functions
Pages 439-440 #1,2,4-7,10,11
Website Video Clips with Solutions:
5.4 Solving Rational Equations
Graphs of rational functions examples
Graphs of rational functions examples of graphing
Watch Concept and Problem #1-3 videos
Graphing rational function video examples 2
Go to topic “algebra” and click on “rational expressions” then click on “graphing rational functions”
Pages 443 #1-3
Solving rational equations examples
Website Video Clips with Solutions:
Solving rational equations example 2
Watch Concept and Problem #1-3 videos
Multiplying and dividing rational expressions review
Watch Concept and Problem #1-3 videos
Adding and subtracting rational expressions review
Watch Concept and Problem #1-3 videos
5.5 Solving Rational Inequalities
Pages 452-453 # 4-8,9,10,14-16
Website Video Clips with Solutions:
Solving rational inequality examples
Solving linear inequalities example 2
5.6 Rates of Change in Rational
Functions
Pages 459-461 #2-10,12,14-18
Website Video Clips with Solutions:
Rates of change in rational functions examples
Review
Pages 468-469
ADVANCED FUNCTIONS MHF4U UNIT 6: Trigonometric Functions
6.1 Radian Measure
Website Video Clips with Solutions:
Pages 320-322
Radians and Degree explanation
Radian and Degree example 1
Go to topic “trigonometry” and click on “radian measure” then click on “radians and degrees”
Radian and Degree Examples 2 to 6
Watch Concept and Problem #1-4 video
Khan Academy Exercises:
6.2 Radian Measure and Angles on
the Cartesian Plane
Website Video Clips with Solutions:
6.3 Graphs of Primary
Trigonometric Functions
Website Video Clips with Solutions:
6.4 Transformations of
Trigonometric Functions
Website Video Clips with Solutions:
Degrees to Radians practice
Radians to Degrees Practice
Degrees and Radians practice
Pages 330-332
Radians and Degree explanation
Finding angles example 1
Finding angles using trig example 2
Pages 336
Finding coordinates on graphs example
Watch Concept and Problem #1-2 video
Review of Sine and Cosine Graph from grade 11
Basic Sine and Cosine Graphs
Go to topic “Trigonometry” and click on “Graphing and Inverse functions” then click on “Basic
Graphs”
Sine and Cosine Graphs
Watch the concept video and the problems #1 to 3
Pages 344-345#
Transformations of Sinusoidal Graphs
Click on topic “Trig Functions” – E: Graphing 15 to 10 – watch and work through the examples of
transformations
More Transformations of Sine and Cosine Graphical examples
Watch the concept video and the problems #1 to 3
Transformations of Sine and Cosine Graph example 2
Watch the concept video and the problems #1 to 3
6.5 Graphs of the Reciprocal
Trigonometric Functions
Website Video Clips with Solutions:
Pages 353 #
Reciprocal Trigonometric functions explanation
Watch the concept video
Reciprocal Trigonometric Graphs
Watch the concept video and the problems #1 to 3
Reciprocal Trig example 1
Go to topic “Trigonometry” and click on “Graphing and Inverse functions” then click on “inverse
trigonometric functions”
6.6 Modelling Trigonometric
Functions
Pages 360-362 #
Website Video Clips with Solutions:
Modelling with Trigonometric Functions examples
6.7 Rates of Change in Trigonometric
Functions
Pages 370-372 #
Website Video Clips with Solutions:
Rate of change in Trigonometric Functions
Review
Pages 374-377 #
ADVANCED FUNCTIONS MHF4U UNIT 7: Trigonometric Identities and Equations
7.1 Equivalent Trigonometric Functions
Pages 392-393 #
Website Video Clips with Solutions:
Solving Trigonometric equations examples
Watch Concept and Problems #1-3 videos
7.2 Compound Angle Formula
Website Video Clips with Solutions:
Khan Academy Exercises:
7.3 Double Angle Formula
Website Video Clips with Solutions:
7.4 Proving Trigonometric Identities
Pages 10-11 #
Cosine Addition formula
Watch Concept and Problems #1-3 videos
Sine Addition formula
Watch Concept and Problems #1-3 videos
Compound Angle Formula examples
Review of trig identities and ratios videos
Trig Ratios video
Click on topic “Trigonometry and Geometry” – D: Finding unknown sides and angles: #18 –24
watch and work through the examples
Evaluating Sin and Cosine at Acute angles
Watch the concept video and the problems #1 to 3
Solving Acute Triangles example
Watch the concept video and the problems #1 to 4
Review your skills
Pythagorean Theorem
Trigonometry Practice 1
Trigonometry Practice 2
Pages 10-11 #
Double Angle formula examples
Watch Concept and Problems #1-2 videos
Double angle Formula example 2
Go to topic “Trigonometry” and click on “Identities and formulas” then click on “double angle
formulas
Pages 10-11 #
Using cosine and sine formulas to prove identities
Website Video Clips with Solutions:
Khan Academy Exercises:
7.5 Solving Linear Trigonometric
Equations
Website Video Clips with Solutions:
Watch Concept and Problems #1-3 videos
Proving Trig identities examples
Watch Concept and Problems #1-3 videos
Review of Identities
Basic Trigonometry identities
Advanced Trigonometry identities
Inverse Trig identity practice
Trig identity practice problems
Pages 10-11 #
Solutions to Tangent at an angle
Watch Concept and Problems #1-2 videos
Solving Trigonometric equations example 1
Solving Trigonometric equations example 2
7.6 Solving Quadratic Trigonometric
Equations
Pages 10-11 #
Website Video Clips with Solutions:
Solving Quadratic Trigonometric Equations examples
Review
Pages 10-11 #
ADVANCED FUNCTIONS MHF4U UNIT 8: Exponential and Logarithmic Functions
8.1 Logarithmic Functions
Website Video Clips with Solutions:
8.2 Transformations of Logarithmic
Functions
Website Video Clips with Solutions:
8.3 Evaluating Logarithms
Website Video Clips with Solutions:
8.4 Laws of Logarithms
Website Video Clips with Solutions:
Pages 451 #
Exponential into Logarithmic graph examples
Watch Concept and Problem #1-3 video
Simplifying/Evaluating logarithms examples
Go to topic “algebra” and click on “logarithms” then click on “logarithms are exponents”
Review of graphing Exponential functions
Watch Concept and Problem #1-4 video
Pages 457-458 #
Graphing Logarithmic Functions explanation
Watch Concept and Problem #1-2 video
Graphing of Logarithmic Function questions
Watch Concept and Problem #1-3 video
Pages 466-467 #
Explaining difference between natural and common logarithms
Changing the base of the exponent examples for question #1
Watch Concept and Problem #1-3 video
Solving simple Logarithms
Watch Concept and Problem #1-5 video
Evaluating Logarithms example
Go to topic “algebra” and click on “logarithms” then click on “equations containing logarithms”
Pages 475-476 #
Product Law
Watch Concept and Problem #1-3 video
Quotient Law of Logarithms
Watch Concept and Problem #1-3 video
Power Law of Logarithms
Watch Concept and Problem #1-3 video
Expanding logarithms rules
Watch Concept and Problem #1-3 video
Simplifying logarithms into condensed form
Watch Concept and Problem #1-3 video
8.5 Solving Exponential Equations
Website Video Clips with Solutions:
8.6 Solving Logarithmic Equations
Website Video Clips with Solutions:
8.7 Problems with Exponential and
Logarithmic Functions
Website Video Clips with Solutions:
Pages 485-486 #
Solving exponential equations with the same base
Watch Concept and Problem #1-4 video
Solving Exponential Equations with different bases
Watch Concept and Problem #1-4 video
Video examples of how to change the base of exponents
Go to topic “algebra” and click on “logarithms” then click on “exponential equations change of base”
Pages 491-492 #
Solving logarithmic Equations
Watch Concept and Problem #1-3 video
Solving Logarithmic Equations with multiple logs
Watch Concept and Problem #1-3 video
Pages 499-501 #
Compound Interest Problems finite
Watch Concept and Problem #1-2 video
Compound Interest continuous problems
Watch Concept and Problem #1-3 video
World problems decay and growth examples
Watch Concept and Problem #1-2 video
More word logarithmic word problems
8.8 Rates of Change in Exponential
and Logarithmic Functions
Pages 507-508 #
Website Video Clips with Solutions:
Rates of Change in Logarithmic and exponential function word problems
See videos above
Review
Pages 404-405 #
ADVANCED FUNCTIONS MHF4U UNIT 9: Combinations of Functions
9.1 Combinations of Functions
Pages 481-482 #1-4,6-10,12
Website Video Clips with Solutions:
Combination of Functions examples
9.2 Combining Two Functions:
Sums and Differences
Pages 490-492 #1,2,4-6,7-12,14,17
Website Video Clips with Solutions:
Combining two functions: sum and difference examples
9.3 Combining Two Functions:
Products
Pages 498-499 #1-9,11-14
Website Video Clips with Solutions:
Combining two functions: product examples
9.4 Quotients of Functions
Pages 511-512 #
Website Video Clips with Solutions:
Quotient of functions examples
9.5 Composition of Functions
Pages 520-522 #
Website Video Clips with Solutions:
Composition of Functions examples
Composition of functions example video
Go to topic “algebra” and click on “functions” then click on “composition”
Composition of functions examples
Watch Concept and Problem #1 video
9.6 Solving Equations and
Inequalities
Pages 531-532 #
Website Video Clips with Solutions:
Different techniques for solving equations and inequalities
9.7 Modelling with Functions
Pages 531-532 #
Website Video Clips with Solutions:
Modelling with functions application problem
Review
Pages 534-535 #