Praxis Math 5165 cheat sheet

A rational number can be written as p/q with integers p and q and q ≠ 0; its decimal expansion either terminates or eventually repeats.

Irrational numbers such as √2, π, and e have infinite, non-repeating decimals; the sum of a rational and an irrational number is always irrational.

Complex numbers have the form a + bi with i² = −1; the modulus is |a + bi| = √(a² + b²) and (a + bi)(a − bi) = a² + b².

Exponent rules: aᵐ·aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, and a^(m/n) = ⁿ√(aᵐ) for a > 0.

To rationalize a denominator containing √a, multiply numerator and denominator by √a; for a denominator a + √b, multiply by the conjugate a − √b.

Scientific notation writes a number as a × 10ⁿ with 1 ≤ |a| < 10; to multiply, multiply the coefficients and add the exponents.

For any two positive integers, GCD(a, b) × LCM(a, b) = a × b.

|x| is the distance from x to 0 on the number line, and the equation |x| = k has the two solutions x = k and x = −k whenever k ≥ 0.

Unit conversion is done by dimensional analysis: multiply by ratios equal to 1 so that unwanted units cancel and only the desired unit remains.

Slope m = (y₂ − y₁)/(x₂ − x₁); point-slope form is y − y₁ = m(x − x₁); parallel lines share a slope and perpendicular slopes multiply to −1.

The quadratic formula gives x = (−b ± √(b² − 4ac))/(2a) for ax² + bx + c = 0 when a ≠ 0.

The discriminant b² − 4ac indicates two distinct real roots when positive, exactly one real root when zero, and two complex roots when negative.

For y = ax² + bx + c the vertex lies at x = −b/(2a); vertex form is y = a(x − h)² + k with vertex (h, k).

Remainder Theorem: dividing p(x) by x − c leaves remainder p(c), so c is a zero of p exactly when p(c) = 0.

Logarithm rules: log(ab) = log a + log b, log(a/b) = log a − log b, log(aᵖ) = p log a, and log_b x = (ln x)/(ln b).

Exponential models A(t) = A₀(1 + r)ᵗ or A(t) = A₀e^(kt) describe growth when r > 0 or k > 0 and decay when those quantities are negative.

Composition is (f ∘ g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, and their graphs reflect across y = x.

Transformations: f(x − h) shifts the graph right h units, f(x) + k shifts it up k units, −f(x) reflects over the x-axis, and af(x) stretches vertically by |a|.

Pythagorean theorem: a² + b² = c² in a right triangle with legs a and b and hypotenuse c.

The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²); the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

A circle with center (h, k) and radius r has equation (x − h)² + (y − k)² = r².

Area formulas: triangle ½bh, circle πr², trapezoid ½(b₁ + b₂)h; volume formulas: cylinder πr²h, sphere (4/3)πr³, cone (1/3)πr²h.

Similar figures have proportional corresponding sides; if the scale factor is k, areas scale by k² and volumes by k³.

Special right triangles: 45-45-90 has sides x, x, x√2; 30-60-90 has sides x, x√3, 2x.

Law of Sines: a/sin A = b/sin B = c/sin C; Law of Cosines: c² = a² + b² − 2ab·cos C.

Radian-degree conversion uses 180° = π radians; arc length is s = rθ and sector area is ½r²θ with θ in radians.

On the unit circle the point at angle θ is (cos θ, sin θ), giving the identity sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ.

Mean = sum of values divided by count; median = middle value of ordered data; mode = most frequent value, and skewness pulls the mean toward the longer tail.

Standard deviation measures the typical distance of data values from the mean; in a normal distribution about 68% of values fall within one standard deviation, 95% within two, and 99.7% within three.

A z-score z = (x − μ)/σ expresses how many standard deviations a value lies from the mean.

Probability rules: P(A or B) = P(A) + P(B) − P(A and B) and P(A|B) = P(A and B)/P(B); events A and B are independent exactly when P(A and B) = P(A)P(B).

Permutations nPr = n!/(n − r)! count ordered arrangements, while combinations nCr = n!/(r!(n − r)!) count unordered selections.

Binomial probability is P(X = k) = C(n, k)pᵏ(1 − p)^(n − k) for n independent trials with success probability p; expected value is Σ x·P(x).

A two-sided limit exists exactly when the left-hand and right-hand limits agree, and f is continuous at c exactly when lim(x→c) f(x) = f(c).

Derivative rules include the power rule d/dx xⁿ = nxⁿ⁻¹, the product rule (uv)′ = u′v + uv′, and the chain rule d/dx f(g(x)) = f′(g(x))g′(x).

The Fundamental Theorem of Calculus states that ∫ₐᵇ f(x) dx = F(b) − F(a) whenever F is an antiderivative of f, so the definite integral gives net signed area.