BQA Grade 12 cheat sheet

Rational numbers are quotients of integers a/b with b≠0; irrational numbers cannot be written in this form and have non-repeating, non-terminating decimal expansions.

The imaginary unit i satisfies i² = −1; a complex number a+bi has real part a and imaginary part b, and its conjugate is a−bi.

For complex numbers, (a+bi)(c+di) = (ac−bd) + (ad+bc)i; the modulus |a+bi| = √(a²+b²).

Exponent rules: a^m · a^n = a^(m+n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, and a^(m/n) = ⁿ√(a^m) for a ≥ 0 when n is even.

Logarithm definition: log_b(x) = y means b^y = x; log_b(xy) = log_b x + log_b y and log_b(x/y) = log_b x − log_b y.

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

Vector addition is componentwise: (a,b) + (c,d) = (a+c, b+d); scalar multiplication k(a,b) = (ka, kb).

The magnitude of vector (a,b) is √(a²+b²); a unit vector in the same direction is (a,b) divided by that magnitude.

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = [−b ± √(b² − 4ac)] / (2a).

The discriminant b² − 4ac determines the number of real roots: positive gives two, zero gives one, negative gives none.

Factoring a difference of squares: a² − b² = (a − b)(a + b); a perfect square trinomial a² + 2ab + b² = (a + b)².

When solving an equation, multiplying or dividing both sides by an expression that could be zero may introduce extraneous solutions, so check answers in the original equation.

For a system of two linear equations, the graphs are parallel (no solution) when slopes are equal and intercepts differ, and identical (infinitely many solutions) when both slope and intercept match.

Exponential growth/decay model: A(t) = A₀(1 + r)^t, where r is the rate per period and t is the number of periods.

The vertex of y = ax² + bx + c occurs at x = −b/(2a); the axis of symmetry is that vertical line.

Polynomial remainder theorem: dividing P(x) by (x − k) leaves remainder P(k); (x − k) is a factor exactly when P(k) = 0.

A function assigns exactly one output to each input in its domain; the vertical line test identifies graphs of functions.

Function notation: f(g(x)) means apply g first, then f; the domain of the composition excludes inputs where g(x) is outside f's domain.

The inverse of a function swaps input and output; its graph is the reflection of the original graph across the line y = x.

Transformations: y = f(x − h) shifts right by h; y = f(x) + k shifts up by k; y = −f(x) reflects across the x-axis.

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis; odd functions satisfy f(−x) = −f(x) and are symmetric about the origin.

Average rate of change of f on [a, b] is [f(b) − f(a)] / (b − a), the slope of the secant line between the two points.

Exponential functions f(x) = a·b^x have constant ratios of consecutive outputs; linear functions have constant differences.

The domain of a rational function excludes values making the denominator zero; those values often produce vertical asymptotes.

Distance formula between (x₁, y₁) and (x₂, y₂): √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).

Slope of a line through two points: m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes and perpendicular lines have slopes whose product is −1.

Area formulas: triangle A = ½bh; circle A = πr²; trapezoid A = ½(b₁ + b₂)h.

Volume formulas: cylinder V = πr²h; cone V = ⅓πr²h; sphere V = (4/3)πr³.

In a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Pythagorean identity: sin²θ + cos²θ = 1; the Law of Sines states a/sin A = b/sin B = c/sin C.

A dilation with scale factor k multiplies lengths by k and areas by k²; a similarity transformation preserves angle measures.

Inscribed angle theorem: an angle inscribed in a circle measures half the intercepted arc; a central angle measures the full arc.

Mean = sum of values divided by count; median is the middle value when data are ordered; mode is the most frequent value.

Standard deviation measures typical distance from the mean; a larger standard deviation indicates more spread.

For independent events A and B, P(A and B) = P(A)·P(B); for mutually exclusive events, P(A or B) = P(A) + P(B).

Conditional probability: P(A | B) = P(A and B) / P(B), provided P(B) > 0.

The complement rule states P(not A) = 1 − P(A).

In a normal distribution, about 68% of data lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

A random sample is more likely to represent a population than a convenience sample; larger samples generally reduce sampling variability.

Correlation measures the strength and direction of a linear association; correlation does not imply causation.

Identify the quantities in a problem, assign variables, and state units before writing equations.

Linear models have a constant rate of change; exponential models have a constant percent change per period.

When fitting a model to data, check whether residuals show a pattern; a patterned residual plot suggests the model form is wrong.

Constraints in a modeling problem (nonnegative lengths, integer counts, capacity limits) restrict the domain of the model.

Dimensional analysis: multiply by conversion factors written as fractions equal to 1 so unwanted units cancel.

For optimization with a linear objective and linear constraints, the optimum occurs at a vertex of the feasible region.

Model predictions outside the range of observed data are extrapolations and carry greater uncertainty than interpolations.

A model is useful if it predicts reasonably, not if it is perfectly accurate; state assumptions explicitly.

A claim is supported by evidence only when the evidence is relevant and sufficient; anecdote alone is weak support.

A thesis statement presents the central argument of a text and is typically found in the introduction.

Transitions such as however, therefore, and consequently signal logical relationships between ideas.

Parallel structure requires items in a list or comparison to share the same grammatical form.

A pronoun must agree with its antecedent in number and gender; ambiguous antecedents create unclear writing.

When summarizing, retain the main idea and key supporting points while omitting minor details and examples.

An author's tone is conveyed through word choice and sentence structure; tone differs from the author's purpose.

Citing a source requires identifying the author and location so a reader can verify the claim.

An independent variable is manipulated; a dependent variable is measured; controlled variables are held constant.

A hypothesis must be testable and falsifiable; a prediction states an expected outcome under specific conditions.

Correlation between two variables does not establish that one causes the other; confounding variables may explain the association.

A control group provides a baseline for comparison; random assignment reduces bias between treatment groups.

When reading a graph, check axis labels, units, and scale; a truncated axis can exaggerate differences.

Significant figures reflect measurement precision; in multiplication and division, the result keeps the least number of significant figures.

Replication and peer review increase confidence in scientific findings; a single study is rarely conclusive.

Outliers should be investigated, not automatically deleted; they may indicate measurement error or a real effect.