MTTC Mathematics (Secondary) (022) cheat sheet

Direct proof derives the conclusion from definitions and given hypotheses; contrapositive proof proves 'not Q implies not P' to establish 'P implies Q'; contradiction assumes the negation of the conclusion.

Mathematical induction: prove a base case, assume the statement for n, and derive it for n+1; strong induction assumes it for all integers from the base through n.

Divisibility tests: by 3 or 9, use the digit sum; by 4, use the last two digits; by 8, use the last three digits; by 11, use the alternating digit sum.

For positive integers a and b, gcd(a,b) = gcd(b, a mod b); also lcm(a,b) = ab / gcd(a,b).

If n = product p_i^{e_i}, the number of positive divisors is product (e_i + 1), and the sum of divisors is product (p_i^{e_i+1} - 1)/(p_i - 1).

A congruence ax = b (mod m) is solvable iff gcd(a,m) divides b; it has gcd(a,m) solutions modulo m when solvable.

Fermat's little theorem: if p is prime and p does not divide a, then a^{p-1} = 1 (mod p); Wilson's theorem: (p-1)! = -1 (mod p) for prime p.

For n a positive integer, sqrt(n) is rational exactly when n is a perfect square; the rational root theorem limits possible rational roots to factors of the constant over factors of the leading coefficient.

Complex numbers: i^2 = -1; (a+bi)(c+di) = (ac-bd) + (ad+bc)i; division uses the conjugate a-bi, and |a+bi| = sqrt(a^2+b^2).

Quadratic formula: x = (-b plus or minus sqrt(b^2 - 4ac))/(2a); discriminant b^2 - 4ac gives two real, one real, or two complex roots according as it is positive, zero, or negative.

Vertex form y = a(x-h)^2 + k has vertex (h,k); for y = ax^2 + bx + c, the axis is x = -b/(2a) and the vertex y-coordinate is f(-b/(2a)).

Function transformations: f(x-h) shifts right h, f(x)+k shifts up k; -f(x) reflects across the x-axis, f(-x) reflects across the y-axis, and a f(x) scales vertically by a.

Exponential model y = a b^t has growth if b > 1 and decay if 0 < b < 1; continuous model y = a e^{kt} has growth if k > 0 and decay if k < 0.

Logarithm rules: log_b(xy) = log_b x + log_b y; log_b(x/y) = log_b x - log_b y; log_b(x^r) = r log_b x; log_b x = ln x / ln b.

Polynomial end behavior follows the leading term: even degree with positive coefficient goes up on both ends; odd degree goes in opposite directions; zeros of even multiplicity touch the axis, odd multiplicity cross it.

Rational function asymptotes: vertical asymptotes occur at noncanceled zeros of the denominator; horizontal asymptote depends on degrees of numerator and denominator; a canceled factor creates a hole.

Sequences: arithmetic a_n = a_1 + (n-1)d with sum n(a_1+a_n)/2; geometric a_n = a_1 r^{n-1} with sum a_1(1-r^n)/(1-r) for r not equal to 1.

Inverse functions: f^{-1} exists iff f is one-to-one; f^{-1}(f(x)) = x, the graphs reflect across y = x, and domain and range swap.

Circle formulas: circumference C = 2 pi r; area A = pi r^2; arc length s = r theta and sector area A = (1/2) r^2 theta when theta is in radians.

Triangle area: A = (1/2)bh; Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) with semiperimeter s = (a+b+c)/2.

Volume formulas: prism or cylinder V = Bh; pyramid or cone V = (1/3)Bh; sphere V = (4/3) pi r^3; sphere surface area S = 4 pi r^2.

Similar figures with linear scale factor k: corresponding lengths scale by k, areas by k^2, and volumes by k^3.

Coordinate geometry: distance = sqrt((x2-x1)^2 + (y2-y1)^2); midpoint = ((x1+x2)/2, (y1+y2)/2); slope = (y2-y1)/(x2-x1).

Parallel lines have equal slopes; perpendicular nonvertical lines have slopes whose product is -1; a horizontal line has slope 0 and a vertical line has undefined slope.

Polygon angles: interior sum = (n-2)180 degrees; exterior sum = 360 degrees; each interior angle of a regular n-gon is (n-2)180/n degrees.

Pythagorean theorem: a^2 + b^2 = c^2 for a right triangle; 45-45-90 sides are x, x, x sqrt(2); 30-60-90 sides are x, x sqrt(3), 2x.

Rigid motions (translations, rotations, reflections) preserve distance and angle measure and produce congruent figures; dilations preserve shape but scale lengths, producing similar figures.

Mean is sensitive to outliers; median and mode are resistant; standard deviation measures typical distance from the mean, and variance is its square.

Normal model: about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3; z = (x - mu)/sigma.

Probability addition rule: P(A or B) = P(A) + P(B) - P(A and B); for independent events P(A and B) = P(A)P(B).

Conditional probability: P(A|B) = P(A and B)/P(B) for P(B) > 0; events are independent exactly when P(A|B) = P(A).

Counting: permutations nPr = n!/(n-r)!; combinations nCr = n!/(r!(n-r)!); the multiplication principle multiplies choices at successive stages.

Binomial probability: P(X = k) = C(n,k) p^k (1-p)^{n-k} for n independent trials with success probability p; mean = np and variance = np(1-p).

Discrete expected value: E(X) = sum x P(x); variance is E(X^2) - [E(X)]^2; the expected value need not be an attainable value.

Graph theory: a connected graph has an Euler circuit exactly when all vertices have even degree and an Euler path exactly when exactly zero or two vertices have odd degree; a tree with n vertices has n-1 edges.

Correlation measures linear association and does not imply causation; the least-squares regression line passes through the point of means and minimizes the sum of squared residuals.

Radians and degrees: 180 degrees = pi radians; arc length s = r theta and sector area A = (1/2)r^2 theta use radians.

Unit-circle identities: sin^2 theta + cos^2 theta = 1; tan theta = sin theta/cos theta; 1 + tan^2 theta = sec^2 theta; 1 + cot^2 theta = csc^2 theta.

Double-angle identities: sin 2theta = 2 sin theta cos theta; cos 2theta = cos^2 theta - sin^2 theta = 1 - 2 sin^2 theta = 2 cos^2 theta - 1.

Law of sines: a/sin A = b/sin B = c/sin C = 2R; law of cosines: c^2 = a^2 + b^2 - 2ab cos C.

For y = A sin(B(x - C)) + D, amplitude = |A|, period = 2pi/|B|, phase shift = C, and vertical shift = D.

Inverse trig ranges: arcsin has range [-pi/2, pi/2]; arccos has range [0, pi]; arctan has range (-pi/2, pi/2).

Limit and continuity: a function is continuous at a if the limit equals the function value; L'Hopital's rule applies to 0/0 or infinity/infinity indeterminate forms.

Derivative rules: d/dx x^n = n x^{n-1}; product, quotient, and chain rules; derivative is instantaneous rate and slope of the tangent line.

Fundamental Theorem of Calculus: if F' = f, the definite integral from a to b of f(x) dx equals F(b) - F(a); critical points occur where f' = 0 or is undefined, and f'' determines concavity.