MHF4U Exam Review: The 10 Question Types That Always Show Up

Every MHF4U final exam in Ontario is written by a different teacher, and every one of them still looks much the same. That is not an accident. The Advanced Functions course has four strands and a short list of expectations in each, so the same ten question types turn up year after year, in January and in June. If you can do these ten cold, you have covered the large majority of the marks on the paper. This review walks through each type: what the question asks, the exact place students lose marks, and how to practise it in the two or three weeks before the exam.

Why the same questions keep showing up

MHF4U, Grade 12 Advanced Functions, is a one-credit university preparation course set out in the Ontario mathematics curriculum for Grades 11 and 12. The curriculum organises it into four strands: Exponential and Logarithmic Functions, Trigonometric Functions, Polynomial and Rational Functions, and Characteristics of Functions. Each strand lists specific expectations, and a final exam has to sample them. Teachers write their own papers, but with only four strands to draw from and a limited number of skills that fit a two-hour sitting, the questions converge. Ten types cover almost all of it.

The ten types below are listed in the order most schools teach them, which is also a sensible review order: the algebra from the polynomial strand carries everything after it. For the full course outline, including how a weekly lesson runs, see our MHF4U tutoring page.

The 10 question types

1. Factor a cubic or quartic completely and solve the equation

You are given a polynomial of degree 3 or 4 and asked to factor it fully, then find all real zeros, or to solve the matching equation. The expected route is the factor theorem: test factors of the constant term, find one zero, divide out the linear factor with long or synthetic division, and factor what remains. Where marks go missing: stopping after the first division when the remaining quadratic still factors, or forgetting to test negative and fractional values. A quick check: the number of zeros, counted with multiplicity, must equal the degree.

2. Sketch a polynomial from its equation, or write the equation from a graph

In one direction you are given a factored polynomial and asked to sketch it, showing intercepts, end behaviour and the shape at each zero. In the other you are given a graph and asked for the equation, including the leading coefficient. Where marks go missing: treating a repeated zero like a single one. A zero of order 2 touches the axis and turns back; a zero of order 3 flattens as it passes through. In the reverse direction, students write the factors correctly and then forget to use the given point to find the vertical stretch.

3. Analyse a rational function: asymptotes, holes and the sketch

Given a rational function, state the domain, find the vertical and horizontal asymptotes and any hole, then sketch. This is the single most common place marks go missing in the first half of the course. Students memorise "vertical asymptote where the denominator is zero" and then draw an asymptote where there is actually a hole, or forget that the horizontal asymptote changes when the degrees of numerator and denominator are equal. The fix: factor first, cancel any common factor (that is the hole), then compare degrees before drawing anything. Practise it on the free asymptotes of rational functions set.

4. Solve a polynomial or rational inequality

Solve something like a factored cubic greater than zero, or a rational expression less than or equal to a constant, and give the answer in interval notation. The method is the same for both: get zero on one side, find the critical values from the zeros and the restrictions, build a sign chart, and read off the intervals. Where marks go missing: multiplying both sides of a rational inequality by an expression that might be negative, which flips the inequality for some values and not others, and including a restricted value in the final interval. Our polynomial inequalities practice page has worked examples.

5. Simplify or evaluate using the laws of logarithms

Expand a single logarithm into a sum and difference, condense several into one, or evaluate an expression exactly without a calculator. There are only three laws, but students combine them into things that are not laws at all. The most common invented rule splits the logarithm of a sum into two logarithms. The fix: write each law beside its exponent-law twin, and test any step you are unsure of with a quick numerical example such as log base 2 of 8. The laws of logarithms set is the one our tutors assign most.

6. Solve an exponential or logarithmic equation and check for extraneous roots

Two versions appear on almost every paper. The first is an exponential equation where the bases can be matched, or where you take a logarithm of both sides. The second is a logarithmic equation that, after you combine the logs and switch to exponential form, becomes a quadratic. Where marks go missing: the quadratic gives two answers, and one of them makes the argument of the original logarithm negative or zero. Students who do not substitute back lose a mark on a question they otherwise solved. Write "check" as the last line of your method before you start.

7. An exponential growth or decay application

Half-life, doubling time, compound interest, population growth, pH or the decibel scale. You are given a situation, asked to write the model, and then asked either for a value at a given time or for the time it takes to reach a given value. The second part always needs a logarithm. Where marks go missing: setting up the exponent incorrectly, usually by dividing the time by the wrong period, and rounding too early. Keep the exact expression until the last line and state the units in your conclusion.

8. Exact trigonometric values in radians, and the graph of a transformed trig function

Evaluate sine, cosine or tangent, or a reciprocal ratio, at an angle given in radians, exactly, with no calculator. Then, or in a separate question, state the amplitude, period, phase shift and vertical shift of a transformed function and sketch one or two cycles. Where marks go missing: students who still think in degrees translate every value, and the translation slips under time pressure. Build the habit of reading pi over 2, pi over 3 and pi over 4 as positions on the unit circle, not as conversions. The exact values at special angles set is built for this.

9. Prove a trigonometric identity

Prove that two expressions are equal for all values where both are defined, using the Pythagorean, quotient, reciprocal, compound angle and double angle identities. Proofs feel like guessing to students who have no strategy. The strategy: start from the more complicated side, convert everything to sine and cosine, look for a Pythagorean identity in disguise, and work on one side only unless your teacher allows meeting in the middle. State the restrictions and finish with a line that says the two sides are equal. That last line is a communication mark.

10. Solve a trigonometric equation on a given interval, then a rates of change question

Two parts that often sit on the same page. First, solve a trigonometric equation over an interval such as zero to two pi, including one that factors as a quadratic in sine or cosine. Second, from the Characteristics of Functions strand, find the average rate of change over an interval and estimate the instantaneous rate of change at a point. Where marks go missing: giving only the first solution when the interval holds two or four, and confusing the two kinds of rate of change. Average uses two points; instantaneous squeezes the interval toward one. Practise both on the solving trigonometric equations and rate of change pages.

Where the marks actually go missing

Across the ten types, the pattern is the same one we see in MCV4U: the new idea is usually clear, and the marks go missing in the algebra underneath it. Factoring that did not fully settle in MCR3U, a dropped negative in a division, an exponent law applied backwards. Spend one evening on Grade 11 algebra before you review anything new.

The second leak is communication. Ontario assessment rewards reasoning, not only answers. Stating the domain, labelling the asymptotes on a sketch, writing a concluding sentence with units, and justifying each step of an identity proof are each worth real marks. On an exam, nothing is obvious until it is written down.

The third leak is the check. Rational and logarithmic equations produce answers that do not work in the original equation, and trig equations on an interval produce more answers than the first one you find. A fixed last step, "check each answer against the original and the interval", recovers marks on two or three questions per paper.

A two-week MHF4U exam review plan

1. Days 1 and 2, polynomials: types 1, 2 and 4. Redo every polynomial question from your unit test, then a fresh set from your textbook's review section. Start a one-page mistake log: the error and the reason behind it.
2. Day 3, rationals: type 3 and the rational half of type 4. Factor first, every time.
3. Days 4 and 5, logarithms: types 5, 6 and 7. Write the three log laws and their exponent twins at the top of the page before you begin, and write "check" at the bottom.
4. Days 6 to 8, trigonometry: types 8, 9 and 10. One day on exact values and graphs, one on identities (five or six proofs per day), one on equations over an interval.
5. Day 9, rates of change and combining functions: the second half of type 10, plus sums, products and compositions of functions with their domains.
6. Days 10 to 12, full papers: past tests from your own class, under a timer, mixed strands. Mark them, read the mistake log, redo anything you missed.
7. Days 13 and 14: light review of the mistake log, the identity page and the unit circle. Sleep.

If you want the review run for you, our exam and test prep tutoring follows the same order: a short diagnostic to find which of the ten types are shaky, targeted lessons on those, then full past papers under a timer. Lessons are $40 per hour 1-on-1, $25 per student per hour in a small group of 2 to 3 (classmates reviewing for the same exam often do this), or $35 per hour each when your family has more than one 1-on-1 class in a week. Custom practice material is included, and the first 30-minute trial is free with no commitment. Our tutors are high-achieving tutors who took MHF4U recently. If MCV4U is next semester, the MCV4U course guide explains what to expect, and the high school math tutoring guide covers the whole Grade 9 to 12 path.

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