flagOntario

Grade 12

Master Grade 12 Advanced Functions

Watch MHF4U video lessons, practice with university-prep problems

students image

Logarithms were impossible until StudyPug. The step-by-step videos made everything click. Went from C+ to A- in one semester.

Alex Kim

pug

Why Grade 12 Advanced Functions Students Choose StudyPug

The complete learning system to master MHF4U

Search with Photo

Search with Photo

Snap a photo of any problem and get the exact lesson

Expert Video Teaching

Expert Video Teaching

Certified teachers explain every concept with clear examples

Unlimited Practice

Unlimited Practice

Thousands of practice questions with step-by-step solutions

How Ontario Students Use StudyPug

1

Select Grade Level
Select Grade Level

Choose your Ontario grade (K-12) and current math topics.

2

Get Unstuck
Get Unstuck

Upload homework problems or browse curriculum-aligned lessons.

3

Practice & Master
Practice & Master

Work through similar problems until concepts stick.

4

See Results
See Results

Track progress and watch grades improve week by week.

Find the right Ontario Grade 12 course

Grade 12 Advanced Functions, University Preparation (MHF4U)

Grade 12 Calculus and Vectors, University Preparation (MCV4U)

Grade 12 Mathematics of Data Management, University Preparation (MDM4U)

Grade 12 Mathematics for College Technology, College Preparation (MCT4C)

Grade 12 Foundations for College Mathematics, College Preparation (MAP4C)

Grade 12 Mathematics for Work and Everyday Life, Workplace Preparation (MEL4E)

Ontario MHF4U Help: Master Advanced Functions FastHelp

Print

OE_ID

Expectations

StudyPug Topic

ON.OE.12AF.A1.1

12AF.A1.1: Recognize the logarithm of a number to a given base as the exponent to which the base must be raised to get the number, recognize the operation of finding the logarithm to be the inverse operation of exponentiation, and evaluate simple logarithmic expressions

ON.OE.12AF.A1.2

12AF.A1.2: Determine, with technology, the approximate logarithm of a number to any base, including base 10

ON.OE.12AF.A1.3

12AF.A1.3: Make connections between related logarithmic and exponential equations and solve simple exponential equations by rewriting them in logarithmic form

ON.OE.12AF.A1.4

12AF.A1.4: Make connections between the laws of exponents and the laws of logarithms, verify the laws of logarithms, and use the laws of logarithms to simplify and evaluate numerical expressions

ON.OE.12AF.A2.1

12AF.A2.1: Determine, through investigation with technology, key features of the graphs of logarithmic functions

ON.OE.12AF.A2.2

12AF.A2.2: Recognize the relationship between an exponential function and the corresponding logarithmic function to be that of a function and its inverse, deduce that the graph of a logarithmic function is the reflection of the graph of the corresponding exponential function in the line y = x, and verify the deduction using technology

ON.OE.12AF.A2.3

12AF.A2.3: Determine, through investigation using technology, the roles of the parameters d and c in functions of the form y = log (x ? d) + c and the roles of the parameters a and k in functions of the form y = alog (kx), and describe these roles in terms of transformations on the graph of f(x) = log x

ON.OE.12AF.A2.4

12AF.A2.4: Pose problems based on real-world applications of exponential and logarithmic functions 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

ON.OE.12AF.A3.1

12AF.A3.1: Recognize equivalent algebraic expressions involving logarithms and exponents, and simplify expressions of these types

ON.OE.12AF.A3.3

12AF.A3.3: Solve simple logarithmic equations in one variable algebraically

ON.OE.12AF.B1.1

12AF.B1.1: Recognize the radian as an alternative unit to the degree for angle measurement, define the radian measure of an angle, and develop and apply the relationship between radian and degree measure

ON.OE.12AF.B1.2

12AF.B1.2: Represent radian measure in terms of ? and as a rational number

ON.OE.12AF.B1.3

12AF.B1.3: Determine, with technology, the primary trigonometric ratios and the reciprocal trigonometric ratios of angles expressed in radian measure

ON.OE.12AF.B1.4

12AF.B1.4: Determine, without technology, the exact values of the primary trigonometric ratios and the reciprocal trigonometric ratios for the special angles 0, ?/6, ?/4, ?/3, ?/2, and their multiples less than or equal to 2?

ON.OE.12AF.B2.1

12AF.B2.1: Sketch the graphs of f(x) = sin x and f(x) = cos x for angle measures expressed in radians, and determine and describe some key properties in terms of radians

ON.OE.12AF.B2.2

12AF.B2.2: Make connections between the tangent ratio and the tangent function by using technology to graph the relationship between angles in radians and their tangent ratios and defining this relationship as the function f(x) = tan x, and describe key properties of the tangent function

ON.OE.12AF.B2.3

12AF.B2.3: Graph, with technology and using the primary trigonometric functions, the reciprocal trigonometric functions for angle measures expressed in radians, determine and describe key properties of the reciprocal functions, and recognize notations used to represent the reciprocal functions

ON.OE.12AF.B2.4

12AF.B2.4: Determine the amplitude, period, and phase shift of sinusoidal functions whose equations are given in the form f(x) = a sin (k(x ? d)) + c or f(x) = a cos(k(x ? d)) + c, with angles expressed in radians

ON.OE.12AF.B2.6

12AF.B2.6: Represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians

ON.OE.12AF.B2.7

12AF.B2.7: Pose problems based on applications involving a trigonometric function with domain expressed in radians, and solve these and other such problems by using a given graph or a graph generated with or without technology from a table of values or from its equation

ON.OE.12AF.B3.1

12AF.B3.1: Recognize equivalent trigonometric expressions and verify equivalence using graphing technology

ON.OE.12AF.B3.2

12AF.B3.2: Explore the algebraic development of the compound angle formulas, and use the formulas to determine exact values of trigonometric ratios

ON.OE.12AF.B3.3

12AF.B3.3: Recognize that trigonometric identities are equations that are true for every value in the domain, prove trigonometric identities through the application of reasoning skills, using a variety of relationships, and verify identities using technology

ON.OE.12AF.C1.1

12AF.C1.1: Recognize a polynomial expression and the equation of a polynomial function, give reasons why it is a function, and identify linear and quadratic functions as examples of polynomial functions

ON.OE.12AF.C1.2

12AF.C1.2: Compare, through investigation using graphing technology, the numeric, graphical, and algebraic representations of polynomial functions

ON.OE.12AF.C1.3

12AF.C1.3: Describe key features of the graphs of polynomial functions

ON.OE.12AF.C1.4

12AF.C1.4: Distinguish polynomial functions from sinusoidal and exponential functions, and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions

ON.OE.12AF.C1.5

12AF.C1.5: Make connections, through investigation using graphing technology, between a polynomial function given in factored form and the x-intercepts of its graph, and sketch the graph of a polynomial function given in factored form using its key features

ON.OE.12AF.C1.6

12AF.C1.6: 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 of f(x) = x and f(x) = x?

ON.OE.12AF.C1.8

12AF.C1.8: Determine the equation of the family of polynomial functions with a given set of zeros and of the member of the family that passes through another given point

ON.OE.12AF.C1.9

12AF.C1.9: Determine, through investigation, and compare the properties of even and odd polynomial functions

ON.OE.12AF.C2.1

12AF.C2.1: Determine, through investigation with and without technology, key features of the graphs of rational functions that are the reciprocals of linear and quadratic functions, and make connections between the algebraic and graphical representations of these rational functions

ON.OE.12AF.C2.2

12AF.C2.2: Determine, through investigation with and without technology, key features of the graphs of rational functions that have linear expressions in the numerator and denominator, and make connections between the algebraic and graphical representations of these rational functions

ON.OE.12AF.C2.3

12AF.C2.3: Sketch the graph of a simple rational function using its key features, given the algebraic representation of the function

ON.OE.12AF.C3.1

12AF.C3.1: Make connections, through investigation using technology, between the polynomial function f(x), the divisor x ? a, the remainder from the division f(x)/(x ? a), and f(a) to verify the remainder theorem and the factor theorem

ON.OE.12AF.C3.3

12AF.C3.3: Determine, through investigation using technology, the connection between the real roots of a polynomial equation and the x-intercepts of the graph of the corresponding polynomial function, and describe this connection

ON.OE.12AF.C3.4

12AF.C3.4: Solve polynomial equations in one variable, of degree no higher than four, by selecting and applying strategies, and verify solutions using technology

ON.OE.12AF.C3.5

12AF.C3.5: Determine, through investigation using technology, the connection between the real roots of a rational equation and the x-intercepts of the graph of the corresponding rational function, and describe this connection

ON.OE.12AF.C3.6

12AF.C3.6: Solve simple rational equations in one variable algebraically, and verify solutions using technology

ON.OE.12AF.C3.7

12AF.C3.7: Solve problems involving applications of polynomial and simple rational functions and equations

ON.OE.12AF.C4.1

12AF.C4.1: Explain, for polynomial and simple rational functions, the difference between the solution to an equation in one variable and the solution to an inequality in one variable, and demonstrate that given solutions satisfy an inequality

ON.OE.12AF.C4.2

12AF.C4.2: Determine solutions to polynomial inequalities in one variable and to simple rational inequalities in one variable by graphing the corresponding functions, using graphing technology, and identifying intervals for which x satisfies the inequalities

ON.OE.12AF.C4.3

12AF.C4.3: Solve linear inequalities and factorable polynomial inequalities in one variable in a variety of ways, and represent the solutions on a number line or algebraically

ON.OE.12AF.D1.1

12AF.D1.1: Gather, interpret, and describe information about real-world applications of rates of change, and recognize different ways of representing rates of change

ON.OE.12AF.D1.2

12AF.D1.2: Recognize that the rate of change for a function is a comparison of changes in the dependent variable to changes in the independent variable, and distinguish situations in which the rate of change is zero, constant, or changing by examining applications

ON.OE.12AF.D1.3

12AF.D1.3: Sketch a graph that represents a relationship involving rate of change, as described in words, and verify with technology when possible

ON.OE.12AF.D1.4

12AF.D1.4: Calculate and interpret average rates of change of functions arising from real-world applications, given various representations of the functions

ON.OE.12AF.D1.6

12AF.D1.6: Determine, through investigation using various representations of relationships, approximate instantaneous rates of change arising from real-world applications by using average rates of change and reducing the interval over which the average rate of change is determined

ON.OE.12AF.D1.7

12AF.D1.7: Make connections, through investigation, between the slope of a secant on the graph of a function and the average rate of change of the function over an interval, and between the slope of the tangent to a point on the graph of a function and the instantaneous rate of change of the function at that point

ON.OE.12AF.D1.8

12AF.D1.8: Determine, through investigation using a variety of tools and strategies, the approximate slope of the tangent to a given point on the graph of a function by using the slopes of secants through the given point

ON.OE.12AF.D1.9

12AF.D1.9: Solve problems involving average and instantaneous rates of change, including problems arising from real-world applications, by using numerical and graphical methods

ON.OE.12AF.D2.1

12AF.D2.1: Determine, through investigation using graphing technology, key features of the graphs of functions created by adding, subtracting, multiplying, or dividing functions, and describe factors that affect these properties

ON.OE.12AF.D2.2

12AF.D2.2: Recognize real-world applications of combinations of functions, and solve related problems graphically

ON.OE.12AF.D2.3

12AF.D2.3: Determine, through investigation, and explain some properties of functions formed by adding, subtracting, multiplying, and dividing general functions

ON.OE.12AF.D2.8

12AF.D2.8: Make connections, through investigation using technology, between transformations of simple functions and the composition of these functions with a linear function

ON.OE.12AF.D3.1

12AF.D3.1: Compare, through investigation using a variety of tools and strategies, the characteristics of various functions

ON.OE.12AF.D3.2

12AF.D3.2: Solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques

ON.OE.12AF.D3.3

12AF.D3.3: Solve problems, using a variety of tools and strategies, including problems arising from real-world applications, by reasoning with functions and by applying concepts and procedures involving functions
Complete Ontario Grade 12 Advanced Functions, University Preparation (MHF4U) Coverage

MHF4U Lessons

181

Video Explanations

1008

Practice Problems

1918

Ontario Standards

100% Aligned

Why Ontario Grade 12 Advanced Functions Students Love StudyPug

University Preparation

University Preparation

Master advanced functions for Ontario university admission—be ready for next level

Canadian Certified Teachers

Canadian Certified Teachers

Learn from expert Ontario teachers who know exactly what you need for MHF4U

Learn Anywhere

Learn Anywhere

Desktop, tablet, or phone—your MHF4U lessons sync across all devices

Get Started Today

Success Stories

Polynomial functions made zero sense. The video lessons broke down factoring perfectly. Aced my final and got into Waterloo Engineering.

Jordan Chen

Used StudyPug for all of Advanced Functions. The photo search saved hours on homework. Cut my study time in half and got a 92%.

Taylor Martinez

Read More

Frequently Asked Questions

Everything you need to know about mastering Grade 12 Advanced Functions with StudyPug

What does MHF4U coverage include?

Complete MHF4U curriculum with video lessons on exponential and logarithmic functions, trigonometry, polynomials, and rational functions. Includes practice problems, step-by-step solutions, and progress tracking for every topic.

How does photo search work for Advanced Functions?

Take a photo of any advanced functions problem, and StudyPug finds the exact lesson teaching that concept. It's like having a personal tutor who knows exactly what you need for university preparation.

Are the teachers certified Ontario educators?

Yes! Our teachers are Canadian certified Ontario educators who understand MHF4U curriculum and create lessons specifically for Ontario university preparation standards.

Can I use StudyPug on my phone or tablet?

Absolutely! StudyPug works on desktop, tablet, and mobile. Your progress syncs automatically so you can study for university preparation anywhere.

How will StudyPug help me prepare for university?

We cover every MHF4U topic required for Ontario university admission and teach the exact concepts you need. Students report significantly improved marks and confidence for university-level mathematics.

How StudyPug Works
Get Started

Practice Smart, See Real Progress

Unlimited Targeted Practice
Unlimited Targeted Practice

10,000+ questions adjust to your exact skill level. Never run out of problems that challenge you.

Visual Progress Tracking
Visual Progress Tracking

See mastery percentage for every topic. Parents get weekly progress emails automatically.

Achievement System
Achievement System

Earn badges for consistency and improvement. Build learning streaks that motivate daily practice

Detailed Analytics
Detailed Analytics

Time spent, problems solved, concepts mastered. Identify exactly where more practice is needed.

student
CallToActionContent

End Math Struggles Today

Ontario curriculum-aligned help that actually works

Get Started
mathImage