Precalculus and Algebra 2: the twelve ideas you must own before calculus (with a self-test and the Ontario map)

Precalculus is the set of ideas a first calculus course assumes you already own: function notation and composition, domain and range, transformations, inverses, polynomial behaviour and factoring, rational functions and asymptotes, exponentials and logarithms, trigonometry in radians with the unit circle, the trig identities, and an intuition for limits and rates of change. Algebra 2 covers roughly the first half. Here are the twelve, each with where calculus uses it, a twelve-problem self-test with the answers kept until after you have tried, and the Ontario map for Canadian readers.

What precalculus is, and how it differs from Algebra 2

In most of the United States the sequence runs Algebra 1, Geometry, Algebra 2, Precalculus, Calculus, with the names varying by state and by school: some call the fourth course Trigonometry, some fold it into an "Algebra 3", and some integrated programs spread the same content over three years without the labels. The content is more stable than the names.

Algebra 2 builds the function families and how to solve their equations: linear and quadratic, then polynomial, rational, exponential and logarithmic, with the log laws usually arriving here. Precalculus takes the same families further and adds what Algebra 2 does not reach: trigonometry in radians, the unit circle, the identities, and the first idea of a limit and a rate of change. Some schools add polar coordinates, parametric equations, vectors or sequences and series, which a calculus course will meet later and does not assume on day one.

The Ontario equivalent is the Grade 12 course MHF4U, Advanced Functions, and the MHF4U map covers every idea on this page in teaching order. It is one of the six Ontario Grade 12 U courses.

The twelve ideas calculus assumes on day one

Each idea comes with where calculus uses it, which is why it is on the list. If you cannot see the calculus use yet, that is fine; it is there so that in September you recognise the idea when it comes back.

1. Function notation and evaluation. The derivative is defined with f(x + h) − f(x); if that expression is unfamiliar, the definition is unreadable.
2. Domain and range. A derivative exists only where the function does, and the first thing a curve-sketching question asks is where the function is defined.
3. Composition of functions. The chain rule differentiates a function inside a function, and you cannot apply it to a composite you cannot take apart.
4. Inverse functions. The derivatives of the logarithm and of the inverse trig functions come from their inverses, and a function must be one-to-one to have one.
5. Transformations. Reading a shift, a stretch or a reflection off an equation is how you sketch before you differentiate and how you check an answer against a picture.
6. Polynomial end behaviour and factoring. Limits are evaluated by factoring, and curve sketching starts from the zeros and the end behaviour.
7. Rational functions and asymptotes. Limits at infinity and vertical asymptotes are the same idea as the asymptotes you found in precalculus, with a limit sign in front.
8. Exponentials and logarithms, including the laws. Exponential and logarithmic functions have their own derivative rules, which assume you can rewrite with the log laws, and e becomes the natural base for both.
9. Radians and the unit circle. Every trigonometric derivative formula is true only in radians; in degrees the formulas pick up a constant and break.
10. The graphs of sine, cosine and tangent. Curve sketching and periodic models need the graphs, their periods and their amplitudes read from the equation.
11. The fundamental and compound-angle identities. A trig expression is usually simplified with an identity before it is differentiated, and the two special trig limits are proved with them.
12. Average rate of change and the idea of a limit. The derivative is the limit of the average rate of change as the interval shrinks; the whole course is this one sentence made precise.

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

The self-test: one problem per idea

Twelve problems, one per idea, in the same order. Do every one on paper with nothing open before you read the answers in the next section. Mark each Right, Struggled or Wrong; a Struggled counts as a miss for the score. Under nine of twelve means the calculus course you are about to take will be a precalculus review course in disguise, and the two-week plan below is for you.

1. If f(x) = x² − 3x, find and simplify f(x + h) − f(x).
2. State the domain of f(x) = √(x − 2).
3. If f(x) = 2x + 3 and g(x) = x², find f(g(2)) and g(f(2)).
4. Find the inverse of f(x) = 3x − 5.
5. For y = 2 sin(3(x − π/6)) + 1, give the amplitude, the period, the phase shift and the midline.
6. Factor x³ − 4x completely.
7. Give the horizontal and vertical asymptotes of y = (3x² + 1)/(x² − 4).
8. Solve log x + log(x − 3) = 1 (base 10).
9. Convert 150° to radians and give the exact value of cos(5π/6).
10. State the period of y = tan 2x.
11. Simplify sin²x/(1 − cos x).
12. Find the average rate of change of f(x) = x² from x = 1 to x = 3.

The answers, derived

Read these only after your twelve attempts. Each one is worked, so you can find the line where yours left the path.

1. f(x + h) − f(x) = (x + h)² − 3(x + h) − (x² − 3x) = x² + 2xh + h² − 3x − 3h − x² + 3x = 2xh + h² − 3h. Every term left has an h in it, which is what makes the derivative definition work.
2. The radicand must be at least zero: x − 2 ≥ 0, so x ≥ 2.
3. 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. They are not equal, and composition never promises they will be.
4. Write y = 3x − 5, swap and solve: x = 3y − 5, so y = (x + 5)/3, and f⁻¹(x) = (x + 5)/3. Check: f(f⁻¹(x)) = 3 · (x + 5)/3 − 5 = x.
5. Amplitude 2; period 2π/3; phase shift π/6 to the right; midline y = 1. The period is 2π divided by the 3, and the shift reads off the (x − π/6) after the 3 has been factored out.
6. x³ − 4x = x(x² − 4) = x(x − 2)(x + 2). The common wrong line stops at x(x² − 4), one step short of complete.
7. Degrees equal on top and bottom, so the horizontal asymptote is the ratio of leading coefficients, y = 3. The denominator is zero at x = 2 and x = −2, and the numerator is not, so both are vertical asymptotes.
8. Combine with the product law: log(x(x − 3)) = 1, so x(x − 3) = 10, so x² − 3x − 10 = 0, so (x − 5)(x + 2) = 0. Of x = 5 and x = −2, only x = 5 survives, because log(−2) is undefined. Dropping the check is the common wrong line.
9. 150° × π/180 = 150π/180 = 5π/6. The angle 5π/6 sits in the second quadrant with reference angle π/6, so cos(5π/6) = −√3/2.
10. The period of tan x is π, and the 2 compresses it: π/2.
11. Replace sin²x with 1 − cos²x and factor the difference of squares: (1 − cos x)(1 + cos x)/(1 − cos x) = 1 + cos x, for cos x ≠ 1.
12. (f(3) − f(1))/(3 − 1) = (9 − 1)/2 = 4. In calculus this becomes the slope of the secant, and the derivative at x = 1 is what it tends to as 3 moves toward 1.

The three that fail calculus students most

By where they appear in every calculus course's first month, three of the twelve do the most damage when they are missing. This is said by the structure of a calculus course, not by any statistic.

Composition (idea 3). The chain rule arrives in about the third week and is used in nearly every derivative after it, and it cannot be learned without taking a composite apart into an outer and an inner function. The repair, over two days: a worked example of composing and decomposing, redone cold; then ten composites of your own, each written as f(g(x)) with f and g named.

Radians and the unit circle (idea 9). Every trigonometric derivative assumes radians, and a student who has to look up cos(5π/6) is stopped on every trig question. The repair: the unit circle's exact values as recall cards, question on the front, nothing that gives the answer away, brought back daily until they are automatic. How to write a card that forces recall has the rules for the front.

The log laws (idea 8). Every exponential model, from population growth to radioactive decay, is solved with them, and the derivative of a logarithm of a product is found by rewriting first. The repair: the three laws on cards, then a worked example of each equation type, redone cold, with a sibling.

For all three, the cycle is the one in how to study math: worked example, redo cold, sibling, error log, and the next day's recall. Sweller and Cooper (1985) found that students who studied worked examples before attempting problems learned algebraic procedures faster and with fewer errors than students who only solved problems, which is why the worked example comes first in every repair.

How to study precalculus, and how to review it before calculus

The procedures. Factoring, solving equations of every family, finding inverses and asymptotes are procedures, and the method is the worked-example cycle above, with an error log that records each wrong line beside the right one.

The facts. The unit circle, the exact values, the log laws and the fundamental identities are recall, not procedure. Roediger and Karpicke (2006) had students either reread a passage or recall it with the page closed; a week later the group that recalled remembered more, even though the rereaders had felt more confident. The lesson for the unit circle: test yourself on it with the page closed, and never study it by looking at it.

The identities. Derive each compound-angle formula once from the one before it on paper, until you can rebuild it under pressure, rather than memorizing the list.

Mixed sets. Once three function families are in play, practise them shuffled. Rohrer and Taylor (2007) found that students who practised problems grouped by type did better during practice and worse on the test a week later than students who practised them shuffled, because the shuffled group had practised choosing the method. Mixed sets across function families shows how to shuffle an exercise set by hand.

The two-week review before calculus starts. Day 1: the self-test above, scored honestly. Days 2 to 13: the misses, by the cycle, two ideas a day, with every idea coming back on a second day before the end; the three ideas above go first whether or not they were misses. Day 14: the self-test again, from memory. The second score is the one that tells you whether September will be calculus or review.

The Ontario map, for Canadian readers

Every one of the twelve ideas is in MHF4U, Ontario's Grade 12 Advanced Functions course, and its 26-idea map has them in this order: function notation, composition, inverses and transformations are the characteristics-of-functions strand; polynomial behaviour and factoring, and rational functions and asymptotes, are the first strand; exponentials and logarithms with the laws are the second; radians, the unit circle, the graphs and the identities are the third; and average rate of change is where the characteristics strand points toward calculus. MHF4U adds what the twelve leave out: polynomial inequalities, rational equations and inequalities, the reciprocal trig functions and exponential models with a context. A Canadian reader taking MHF4U can use the self-test above as a mid-year check, and a US reader can use the MHF4U map as the long version of this list.

Where this goes next

A first calculus course assumes all twelve, teaches limits precisely in its first weeks, then the derivative and the integral; first-year calculus: what changes is the guide for that transition. AP Calculus AB assumes the same twelve and covers the derivative and the integral in one school year; the ideas that matter most for it are the same three above, because its first unit is limits and its second is the derivative rules.

The twelve, on one screen

• Function notation and evaluation
• Domain and range
• Composition of functions
• Inverse functions
• Transformations
• Polynomial end behaviour and factoring
• Rational functions and asymptotes
• Exponentials and logarithms, with the laws
• Radians and the unit circle
• The graphs of sine, cosine and tangent
• The fundamental and compound-angle identities
• Average rate of change and the idea of a limit

Three to make automatic before anything else: composition, the unit circle, the log laws. Take the self-test, repair the misses by the worked-example cycle, retake it in two weeks.