MHF4U Advanced Functions: the course map (26 ideas in the order they are taught), the three ideas everything leans on, and how to study it

MHF4U (Advanced Functions) is the Ontario Grade 12 course that turns Grade 11 functions into the tools calculus needs: polynomial and rational functions, exponential and logarithmic functions, trigonometric functions in radians, and the characteristics of functions (rates of change, combinations, inverses, transformations). Our map has 26 ideas in teaching order, and three of them, polynomial division, the log laws and the trig identities, carry most of what follows. Here is the map and how to study it.

What MHF4U is, and who takes it

The code reads as every Ontario code does: MHF is Advanced Functions, 4 is Grade 12, U is university preparation. The Ministry's course description sets the prerequisite as MCR3U, Grade 11 Functions, and the official description lives on the Ministry of Education's curriculum site. That text is Crown copyright, so this page uses our own idea titles and our own one-line descriptions of the strands.

Most students take MHF4U in Grade 12, often alongside MCV4U, which lists MHF4U as a prerequisite that may be taken before it or at the same time. It leads to MCV4U and then to a first-year calculus course, and most math-based university programs name it as a required course. It is one of the six Grade 12 U courses in math and science.

For a reader in the United States, MHF4U is roughly a precalculus course: the same functions, logs and trigonometry in radians, with no calculus. The precalculus guide lists the twelve ideas a calculus course assumes, and every one of them is on the map below.

The course map: 26 ideas in the order they are taught

This is the pack's teaching order, grouped by strand with our one-line description of each strand. It is the order a class usually meets the ideas, not a ranking of importance.

Polynomial and rational functions (ideas 1 to 8): the algebra the rest of the course runs on.

1. Polynomial functions, degree, leading coefficient and end behaviour
2. Graphs of polynomial functions: zeros, multiplicity and turning points
3. Dividing polynomials and the remainder theorem
4. The factor theorem and factoring polynomial expressions
5. Solving polynomial equations
6. Solving polynomial inequalities
7. Rational functions, vertical and horizontal asymptotes
8. Solving rational equations and inequalities

Exponential and logarithmic functions (ideas 9 to 14): growth, decay and the inverse that undoes them.

9. Exponential functions and their graphs
10. Logarithms as the inverse of exponentiation
11. The laws of logarithms
12. Solving exponential equations
13. Solving logarithmic equations
14. Exponential and logarithmic models

Trigonometric functions (ideas 15 to 21): angles in radians, the unit circle, the identities and the equations.

15. Radian measure and arc length
16. Trigonometric functions of any angle and the unit circle
17. Graphs of sine, cosine and tangent and their transformations
18. Reciprocal trigonometric functions
19. Fundamental trigonometric identities
20. Compound-angle and double-angle formulas
21. Solving trigonometric equations

Characteristics of functions (ideas 22 to 26): the ideas that run under every strand and into calculus.

22. Average and instantaneous rates of change
23. Sums, differences, products and quotients of functions
24. Composition of functions
25. Inverse functions
26. Transformations of functions

Chapter 3 of the free playbook is the method for turning this list into the checklist you keep for the semester: one line per idea, marked each time you test yourself on it.

What each idea unlocks

The order above is a chain inside each strand, and knowing the chain tells you where a gap will show up later.

In the polynomial strand, division (idea 3) gives the remainder theorem, which gives the factor theorem (4), which is how every polynomial equation (5) and inequality (6) is solved, and rational functions (7, 8) are two polynomials divided. A student who cannot divide polynomials is stuck from idea 4 onward.

In the exponential strand, the definition of a logarithm (10) gives the log laws (11), and the two equation types (12, 13) are the laws applied in each direction. The models (14) are the equations with a context attached.

In the trig strand, radians (15) and the unit circle (16) give the graphs (17) and the reciprocals (18); the fundamental identities (19) give the compound-angle formulas (20); and solving an equation (21) uses all of them, because most trig equations have to be rewritten with an identity before they can be solved.

The characteristics strand is different: rates of change (22) point forward to the derivative in MCV4U, and the other four run under everything, because every function in the course can be combined, composed, inverted and transformed. Teachers place this strand first, last or threaded through the year, and the method on this page works in any of those orders.

The three ideas everything leans on

By the map's own dependencies, not by any statistic, three ideas carry the most weight. Each comes with the one thing students get wrong and a two-line check.

Polynomial division and the remainder theorem (idea 3). Everything from the factor theorem to rational equations rests on it. The wrong line: dividing by x − a but evaluating at −a. The check: dividing by x − 2 means the remainder is P(2), with the sign flipped from the divisor. The Feynman walk-through of the remainder theorem shows how to explain why that works until it stops feeling like a rule.

The laws of logarithms (idea 11). Both equation types, the models, and SCH4U's pH calculations use them. The wrong line: log(a + b) = log a + log b. The check: the laws are about products, quotients and powers, never sums; log(a + b) has no shortcut. Test it with a = b = 10: log 20 is about 1.30, and log 10 + log 10 is 2.

The fundamental identities (idea 19). Every compound-angle formula and every trig equation past the simplest rewrites something with sin²x + cos²x = 1 or the quotient identities. The wrong line: treating sin²x as (sin x²). The check: sin²x means (sin x)², so sin²x + cos²x = 1 is true for every x you try, and plugging in x = π/6 (1/4 + 3/4 = 1) takes ten seconds.

Worked example: one problem from each strand, with the common wrong line

Each answer below is re-derived, so you can check your own against it line by line.

Polynomials. Find the remainder when x³ − 2x² + 3x − 5 is divided by x − 2. By the remainder theorem the remainder is P(2) = 8 − 8 + 6 − 5 = 1. The wrong line evaluates at −2 instead: P(−2) = −8 − 8 − 6 − 5 = −27, which is the remainder on division by x + 2, a different question.

Logs. Solve 2^(x + 1) = 10. Take log base 2 of both sides: x + 1 = log₂ 10, so x = log₂ 10 − 1. As a decimal, log₂ 10 = log 10 / log 2 ≈ 3.32, so x ≈ 2.32. Check: 2^3.32 ≈ 10. The wrong line treats the exponent as a multiplier and writes x = 10/2 − 1 = 4, and 2⁵ = 32, not 10.

Trig. Solve 2cos²x − 1 = 0 on [0, 2π]. Then cos²x = 1/2, so cos x = ±√2/2. Cosine is +√2/2 at π/4 and 7π/4, and −√2/2 at 3π/4 and 5π/4, so x = π/4, 3π/4, 5π/4, 7π/4. The wrong line forgets the ± on the square root and keeps only the two positive-cosine solutions.

Characteristics. If f(x) = 2x + 3 and g(x) = x², find f(g(2)) and g(f(2)). Inside first: g(2) = 4, so f(g(2)) = f(4) = 11. The other way: f(2) = 7, so g(f(2)) = g(7) = 49. The wrong line assumes the two are equal. Composition is not commutative, and the two numbers are the proof.

How to study MHF4U

The strands are not all the same kind of knowledge, so the method changes by strand type.

The procedures (division, factoring, solving every kind of equation). Read one worked example slowly, cover it, redo it from the question alone, then do a sibling with different numbers, and write every wrong line in an error log with the right line beside it. Sweller and Cooper (1985) found that students who studied worked examples before attempting problems learned algebra procedures faster and made fewer errors than students who only solved problems. How to study math has the full five-step cycle.

The facts (the unit circle, the exact values, the log laws, the identities). These are recall, not procedure. Put each on a card with the question on the front and nothing that gives the answer away, say the answer before you turn it, and bring the misses back the next day.

The identities (compound angles, double angles). Do not memorize them as a list. Derive each once from the one before it, on paper, until you can rebuild it under pressure; the Feynman walk-through linked above is the way to do that.

The weeks before the exam. Mix the strands. Rohrer and Taylor (2007) gave students the same practice problems either grouped by type or shuffled; the grouped students did better in practice and worse on the test a week later, because the shuffled students had practised deciding which method to use. Mixed sets across strands shows how to shuffle an exercise set by hand.

The calendar. Every idea comes back on at least two separate days. A study plan you will keep fits that into a week that also has four other courses in it.

A week-by-week plan for a semester

An 18-week semester in the pack's order. Each week has one recall task (closed book, five minutes, before anything else) and one mixed set (problems from this week and every earlier week, shuffled). Schools order the strands differently: rearrange the rows, keep the method.

Weeks	Ideas	The recall task	The mixed set
1–2	1–4: polynomial functions, graphs, division, the factor theorem	Sketch a polynomial from its factored form	Division and factor-theorem problems, mixed
3–5	5–8: polynomial equations and inequalities, rational functions and equations	State the asymptotes of a rational function from its equation	Equations and inequalities, polynomial and rational together
6–7	9–11: exponential functions, logarithms, the log laws	Write the log laws from memory, with one example each	Rewrite and evaluate problems, both directions
8–9	12–14: exponential and logarithmic equations, the models	Solve one equation of each type cold	Equations and models, mixed with weeks 6–7
10–11	15–18: radians, the unit circle, the graphs, the reciprocals	The unit circle's exact values, from memory	Conversions, exact values and graph transformations
12–14	19–21: identities, compound angles, trig equations	Derive one compound-angle formula	Trig equations needing an identity, mixed with weeks 10–11
15–16	22–26: rates of change, combining, composing, inverting, transforming	Compose two functions both ways, cold	Characteristics problems using functions from every strand
17	Mixed review by strand	Blank-page recall of each strand's ideas	Sets that mix all four strands
18	Exam week	The three load-bearing ideas, once more	The one-week exam plan below

Exam review in one week

Seven days out, take the 26 ideas and sort them in one sitting: one closed-book attempt per idea, a problem or a blank page with the idea's name at the top, marked into Can, Shaky and Cannot. The one-week exam plan runs from there: the Cannot pile first by the worked-example cycle, the Shaky pile by recall and one problem each, a timed practice run two days out, a light review the night before, and sleep.

One rule is specific to this course: the three load-bearing ideas, division, the log laws and the identities, go into the week whether or not they sorted as Can. They turn up inside other strands' questions, where a student who has not seen them for two months loses marks on a problem that was really about something else.

What carries into MCV4U and first year

Four ideas from this map are used on the first day of calculus. Rates of change (22) become the derivative itself, defined as the limit of the average rate. Composition (24) is the structure the chain rule differentiates. Exponentials and logarithms (9 to 11) get their own derivative rules, which assume the log laws. The identities (19, 20) are what you simplify with before differentiating a trig expression. A student who keeps those four current through the summer starts MCV4U on the course's own terms rather than relearning Grade 12 in the first month, and the same four are what a first-year calculus course assumes with no review.

MHF4U on one screen

• Four strands: polynomial and rational functions; exponential and logarithmic functions; trigonometric functions; characteristics of functions.
• Three load-bearing ideas: polynomial division and the remainder theorem; the laws of logarithms; the fundamental trigonometric identities.
• The method: procedures by the worked-example cycle with an error log; facts by recall cards; identities by derivation; mixed sets across strands before the exam.
• The plan: every idea on at least two separate days; the 18-week table above; the one-week triage before the exam.
• What it leads to: MCV4U and first-year calculus, which use rates of change, composition, logs and the identities from the first day.