MHF4U Tutoring: Advanced Functions

MHF4U is Ontario's Grade 12 Advanced Functions course, the one almost every university program with a math requirement asks for. It takes every function family from Grade 11 deeper, adds logarithms and radians, and expects full, justified solutions. Winny Math tutors MHF4U online, 1-on-1 or in small groups, with a weekly lesson matched to your class, custom practice included, and a free 30-minute trial to start. Solo lessons are $40 per hour. Here is what the course covers, where it gets hard, and how we help.

What MHF4U covers

MHF4U is a one-credit university preparation course. Its prerequisite is MCR3U, Grade 11 Functions, or MCT4C, Grade 12 Mathematics for College Technology. The Ontario curriculum for MHF4U organises the course into four strands. Schools split those strands into their own units and teach them in their own order, often polynomials first and trigonometry last, so your teacher's outline may look different, but the content underneath is the same across the province.

Polynomial and rational functions

Students connect the degree, leading coefficient and zeros of a polynomial to the shape of its graph, including end behaviour and the effect of a repeated zero. They divide polynomials, use the remainder and factor theorems to factor cubics and quartics completely, and solve polynomial equations and inequalities both algebraically and from a graph. Rational functions follow: finding vertical and horizontal asymptotes, spotting a hole in the graph, sketching, and solving rational equations and inequalities with the domain restrictions kept in view.

Exponential and logarithmic functions

The logarithm is introduced as the inverse of an exponential function, and students learn to evaluate logarithms, switch between exponential and logarithmic form, and apply the laws of logarithms to expand, condense and simplify expressions. They solve exponential equations by matching bases or by taking a logarithm of both sides, solve logarithmic equations and check for extraneous roots, and graph both families with transformations. Applications cover growth and decay, compound interest, half-life, pH and the decibel scale.

Trigonometric functions

This strand switches the course from degrees to radians. Students convert between the two, find exact values at the special angles around the unit circle, and graph sine, cosine and tangent, along with the reciprocal functions, with transformations expressed in radians. They prove identities using the Pythagorean, quotient, compound angle and double angle identities, and solve trigonometric equations over a given interval, including equations that factor as quadratics in sine or cosine.

Characteristics of functions

The strand that ties the course together and points at MCV4U. Students compare average and instantaneous rates of change, estimate the slope of a tangent from a table or a graph, and analyse the sum, difference, product, quotient and composition of two functions, including their domains. The course ends with modelling problems that ask students to choose the right function family for a situation and defend the choice.

Where students get stuck

These are the sticking points Brett wrote up in our Grade 12 Advanced Functions guide, drawn from the lessons our tutors run every week. The pattern is the same as in MCV4U: the new idea is usually clear, and the marks go missing in the algebra underneath it.

• Asymptotes of rational functions. Students memorise "vertical where the denominator is zero" and then put an asymptote where there is actually a hole, or forget the horizontal asymptote changes when the degrees are equal. The fix is to factor first, cancel, and compare degrees before drawing anything.
• Mixing up the log laws. The three laws are short, but students combine them in ways that are not laws at all, for example splitting the logarithm of a sum, or struggle when the base changes. Writing each law next to its exponent-law twin, and checking a result with a quick numerical example, stops most of these errors.
• Thinking in degrees inside a radians course. Students who still picture 90 degrees have to translate every time, and the translation slips under test pressure. Building the habit of reading pi over 2, pi over 3 and pi over 4 as positions on the unit circle, not as conversions, makes the whole trig strand faster.
• Too many trig identities, no strategy. Proofs feel like guessing. Starting from the more complicated side, converting everything to sine and cosine, and looking for a Pythagorean identity in disguise turns proofs into a procedure.
• Factoring that did not stick in Grade 11. The factor theorem, synthetic or long division, and factoring a quadratic in a hidden variable all sit on top of MCR3U skills. When those are shaky, the polynomial unit becomes a wall, and it is the first unit, so it sets the tone for the course.
• Extraneous solutions. Logarithmic and rational equations produce answers that do not work in the original equation. Students who do not check their solutions against the domain lose marks on questions they otherwise solved correctly.
• Losing communication marks. Ontario assessment rewards reasoning, not only answers. Stating the domain, labelling asymptotes, writing a concluding sentence and justifying each step of an identity proof is worth real marks on every test.

How a weekly MHF4U lesson runs

Most MHF4U students work with us once a week, 1-on-1, online, with a shared whiteboard. Families who want more in the run-up to a test add a second class that week. A typical hour looks like this.

1. Homework check (about 10 minutes). The student brings the week's homework and the questions they could not finish. The tutor works through the stuck ones and notes the type of error, because a dropped negative sign and a misunderstood log law call for different fixes.
2. The current unit (about 30 minutes). The tutor teaches or re-teaches the idea the class is on, matched to the school's order of units, with worked examples on the whiteboard. When the class is ahead of the student, the lesson catches up. When the student is on track, the lesson works slightly ahead so the next class feels familiar.
3. Targeted practice (about 15 minutes). Questions chosen for the exact weak spot, done live so the tutor can see where the thinking goes off. For MHF4U this is where the factor theorem, log equations and exact trig values get drilled until they are automatic.
4. Custom practice for the week (about 5 minutes). Every student leaves with a short practice set built for them, included in the price, plus the matching free sets on Winny Math Learn. The next lesson starts by marking it.

Our tutors are high-achieving tutors who took MHF4U recently and remember where the traps are. Brett, who founded Winny Math, is a Commerce student at Queen's University and still tutors senior math himself. Lessons run by appointment, including evenings and weekends, which matters in Grade 12 when the timetable is already full.

Exam and test prep for MHF4U

In a semestered Ontario school, MHF4U either runs from September to the exam in late January or from February to the exam in June. The January timing is the most common, because many students take Advanced Functions in semester 1 so that MCV4U can follow in semester 2. That makes the MHF4U mark one of the first Grade 12 marks that universities see, and it lands right before early admission decisions. Non-semestered schools spread the course over the full year with a June exam. Either way, the exam is cumulative, and the trigonometry strand, which usually comes last, gets the least review time unless you plan for it.

Our exam and test prep tutoring for MHF4U follows a simple rule: a week before each unit test we switch from the current lesson to mixed practice under a timer and clear anything still shaky. For the final exam we work backwards from the date. Polynomial and rational functions first, because the algebra carries everything else, then exponential and logarithmic functions, then trigonometry, then rates of change and combining functions, then full past tests from the student's own class. Students keep a short mistake log through the semester and read it before every test, which turns a stack of marked quizzes into a study guide.

For students in AP Calculus or IB Math, MHF4U overlaps heavily with the precalculus content those programs assume. Our AP and IB math tutoring page explains the differences. If the course is going well and you want to be ready for calculus, we also use the last few lessons to preview limits and the derivative so the first weeks of MCV4U feel like review.

Practice MHF4U free on Winny Math Learn

Winny Math Learn is our free practice platform for the Ontario curriculum. Every MHF4U skill has its own page with worked, step-by-step solutions, no account needed. These are the sets our tutors assign most often between lessons.

• Factor theorem and factoring a cubic completely: the Grade 11 skill the whole polynomial unit stands on.
• Vertical and horizontal asymptotes of rational functions: the most common place marks go missing in the first half of the course.
• Laws of logarithms: expanding, condensing and simplifying without inventing a fourth law.
• Radian measure and conversion: the habit that makes the trigonometry strand faster.
• Trigonometric identities and solving trigonometric equations: the proof strategy and the interval check, together.

The full list of MHF4U skills, alongside MCV4U and MDM4U, is on the Grade 12 math practice page.

Pricing

Solo 1-on-1 lessons are $40 per hour. Small-group classes are $25 per student per hour, for 2 to 3 students, which suits classmates studying for the same test. When your family has more than one 1-on-1 class a week, each one is $35 per hour. Custom practice material is included with every lesson. Everyone starts with a free 30-minute trial, with no commitment. Full details, including billing and cancellation, are in the pricing and policies section on the homepage, and our guide to math tutoring costs in Ontario shows how that compares with other options.

Looking at the whole Grade 9 to 12 path? Our high school math tutoring guide covers every course code, from MTH1W to the three Grade 12 options, and which ones universities want. If calculus is next, see our MCV4U tutoring page.

Frequently asked questions