Bridge to university · Mathematics

Bridge: algebra and functions for first year

Review exponent and log laws, factoring, rational expressions, transformations, inverses and polynomial division before first-year math.

41 cards6 topicsAbout 15 minutes a roundGrade 12 advanced functions or pre-calculus review before first-year calculus

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About this deck

This deck collects the algebra and function skills that first-year calculus, linear algebra and science courses quietly assume you already have. It covers exponent and logarithm laws, factoring patterns, simplifying and combining rational expressions, function notation and composition, inverse functions, graph transformations, and polynomial division with the remainder and factor theorems. Many cards include a short worked example so you can see each rule applied, not just stated.

It is written for students finishing Grade 12 (Advanced Functions in Ontario, Math 30-1 in Alberta, Pre-Calculus 12 in British Columbia, or the equivalent) and for anyone returning to math after a break. Go through the cards once to sort them into known and shaky, then spend most of your time on the shaky pile. Try each example on paper before flipping the card. The most common mistakes at this level are splitting a log of a sum, cancelling terms instead of factors, forgetting restrictions on rational expressions, reading the sign of a horizontal shift backwards, and confusing an inverse function with a reciprocal. If you can explain why each of those is wrong, you are in good shape for the first weeks of university math.

Before you start: Grade 11 functions or Math 20-1 level algebra; comfort with fractions and order of operations.

All 41 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Exponents and logs

Product rule for exponents: When multiplying powers with the same base, keep the base and add the exponents: a^m * a^n = a^(m+n).Example: x^3 * x^4 = x^7
Quotient rule for exponents: When dividing powers with the same base, keep the base and subtract the exponents: a^m / a^n = a^(m-n), with a not equal to 0.Example: y^9 / y^2 = y^7
Power of a power: Raising a power to another power multiplies the exponents: (a^m)^n = a^(mn).Example: (2^3)^2 = 2^6 = 64
Zero exponent: Any nonzero base to the power 0 equals 1, because a^n / a^n is both 1 and a^(n-n).Example: 7^0 = 1 and (3x)^0 = 1 for x not 0
Negative exponent: a^(-n) = 1/a^n. The minus sign in the exponent means take the reciprocal; it does not make the value negative.Example: 2^(-3) = 1/8, not -8
Rational exponent: a^(m/n) is the nth root of a raised to the power m. Take the root first to keep numbers small.Example: 8^(2/3) = (cube root of 8)^2 = 2^2 = 4
Logarithm (definition): log_b(x) = y means b^y = x. It answers: what exponent on base b gives x? It needs b > 0, b not 1, and x > 0.Example: log_2(32) = 5 because 2^5 = 32
Product law of logarithms: log_b(MN) = log_b(M) + log_b(N): the log of a product splits into a sum of logs.Example: log(4) + log(25) = log(100) = 2
Quotient law of logarithms: log_b(M/N) = log_b(M) - log_b(N): the log of a quotient becomes a difference.Example: log_3(54) - log_3(2) = log_3(27) = 3
Power law of logarithms: log_b(M^k) = k * log_b(M). This is how you bring an unknown exponent down to solve for it.Example: Solve 3^x = 20: x = log(20)/log(3), about 2.727
Change of base formula: log_b(x) = log(x)/log(b), using any convenient base such as 10 or e. Lets a calculator evaluate a log in any base.Example: log_5(40) = log(40)/log(5), about 2.292
Natural log ln(x): The log with base e (about 2.71828). It undoes e^x: ln(e^x) = x, and e^(ln x) = x for x > 0.Example: ln(e^4) = 4
Log trap: log(a + b): No law splits the log of a sum. log(a + b) is not log(a) + log(b); only products, quotients and powers have laws.Example: log(10 + 10) = log(20), about 1.301, but log(10) + log(10) = 2

Factoring

Common factor first: Before trying any pattern, pull out the greatest common factor of every term. It shrinks the problem and is often the step people skip.Example: 6x^3 - 9x^2 = 3x^2(2x - 3)
Difference of squares: a^2 - b^2 = (a - b)(a + b). It works only with subtraction; a^2 + b^2 does not factor over the real numbers.Example: x^2 - 49 = (x - 7)(x + 7)
Factoring x^2 + bx + c: Find two numbers that multiply to c and add to b, then write (x + p)(x + q).Example: x^2 - x - 12 = (x - 4)(x + 3)
Decomposition for ax^2 + bx + c: Find two numbers that multiply to a*c and add to b, split the middle term with them, then group in pairs.Example: 6x^2 + 7x + 2: use 3 and 4, so 3x(2x + 1) + 2(2x + 1) = (3x + 2)(2x + 1)
Sum and difference of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2) and a^3 - b^3 = (a - b)(a^2 + ab + b^2). The quadratic piece does not factor further.Example: x^3 - 8 = (x - 2)(x^2 + 2x + 4)
Quadratic formula: For ax^2 + bx + c = 0, x = (-b plus or minus sqrt(b^2 - 4ac)) / (2a). Use it when factoring is not obvious.Example: x^2 - 2x - 2 = 0 gives x = 1 plus or minus sqrt(3)
Discriminant: The quantity b^2 - 4ac. Positive: two real roots. Zero: one repeated root. Negative: no real roots.Example: x^2 + 4x + 5: 16 - 20 = -4, so no real roots

Rational expressions

Simplifying a rational expression: Factor the top and bottom completely, then cancel common factors. Record restrictions from the original denominator first.Example: (x^2 - 9)/(x^2 + 3x) = (x - 3)/x, with x not 0 and x not -3
Restrictions (non-permissible values): Values of the variable that make any denominator in the original expression equal 0. They stay excluded even after cancelling.Example: For (x + 1)/((x - 2)(x + 5)), exclude 2 and -5
Adding rational expressions: Find the lowest common denominator, rewrite each fraction over it, combine the numerators, then simplify.Example: 1/x + 2/(x + 1) = (x + 1 + 2x)/(x(x + 1)) = (3x + 1)/(x(x + 1))
Dividing rational expressions: Multiply by the reciprocal of the divisor, factor, then cancel. Also exclude values that make the divisor itself equal 0.Example: (x/3) / (x^2/6) = (x/3)(6/x^2) = 2/x
Cancelling trap: Only factors that multiply the entire top and entire bottom may be cancelled, never pieces joined by plus or minus.Example: (x + 6)/6 is not x + 1; it equals x/6 + 1

Functions and inverses

Function: A rule that gives each input exactly one output. On a graph, no vertical line meets it more than once.Example: y = x^2 qualifies; x = y^2 does not
Domain and range: The set of allowed inputs, and the set of outputs actually produced.Example: f(x) = sqrt(x - 2): inputs x >= 2, outputs y >= 0
Composition f(g(x)): Apply g first, then feed its output into f. Order matters, so f(g(x)) and g(f(x)) are usually different.Example: f(x) = x^2, g(x) = x + 1: f(g(x)) = (x + 1)^2 but g(f(x)) = x^2 + 1
Inverse function: A rule that undoes f: if f(a) = b then it sends b back to a. Find it by swapping x and y and solving for y; its graph is f reflected in y = x.Example: f(x) = 3x - 5 gives f^(-1)(x) = (x + 5)/3
One-to-one requirement: Only a rule where each output comes from a single input has an inverse that is also a function. It passes the horizontal line test.Example: y = x^2 needs the domain x >= 0 to have the inverse y = sqrt(x)
f^(-1)(x) versus 1/f(x): The superscript -1 on a function name means the inverse, not the reciprocal. The reciprocal is written 1/f(x) or (f(x))^(-1).Example: For f(x) = 2x: inverse is x/2, reciprocal is 1/(2x)

Transformations

Vertical translation y = f(x) + k: Moves every point up k units when k > 0 and down when k < 0; the shape stays the same.Example: y = x^2 + 3 is the parabola lifted 3 units
Horizontal translation y = f(x - h): Moves the graph right h units when h > 0. The sign feels backwards: f(x + 3) moves the graph left 3.Example: y = (x - 2)^2 has its vertex at (2, 0)
Vertical stretch y = a f(x): Multiplies every y value by a. Size above 1 stretches, between 0 and 1 compresses, and a negative a also reflects in the x axis.Example: y = -2 sqrt(x) is twice as tall and flipped downward
Horizontal stretch y = f(bx): Divides every x value by b, so b > 1 squeezes the graph toward the y axis. A negative b also reflects in the y axis.Example: y = sin(2x) completes a cycle in half the usual width
Order of transformations: For y = a f(b(x - h)) + k, apply stretches and reflections first, then translations. Factor b out before reading h.Example: y = f(2x - 6) = f(2(x - 3)) shifts right 3, not 6

Polynomials

Long division of polynomials: Divide leading terms, multiply, subtract, bring down, and repeat until the remainder has lower degree than the divisor. Write 0 for missing powers.Example: (x^3 - 2x + 4) / (x - 2) = x^2 + 2x + 2, remainder 8
Remainder theorem: When P(x) is divided by x - a, what is left over equals P(a), so you can find it without dividing.Example: P(x) = x^3 - 2x + 4: P(2) = 8, matching the long division
Factor theorem: x - a divides P(x) evenly exactly when P(a) = 0, so testing likely values finds linear factors.Example: P(x) = x^3 - 6x^2 + 11x - 6: P(1) = 0, so x - 1 is a factor
Synthetic division: A shortcut for dividing by x - a using only coefficients: bring down, multiply by a, add, repeat. The last entry is the remainder.Example: Coefficients 1, 0, -2, 4 with a = 2 give 1, 2, 2 and remainder 8
End behaviour of a polynomial: Set by the leading term. Even degree: both ends point the same way. Odd degree: opposite ways. A negative leading coefficient flips them.Example: y = -x^3 + x rises on the left and falls on the right