This chapter teaches nothing new. Every fact below was already taught in full, with worked examples and practice, somewhere earlier in this book. Its only job is to put the relationships you'll want to recall quickly in one place, organized by Part, so a quick review before test day doesn't mean flipping back through thirty chapters.
Remember what Part IX already established: there is no general Math formula sheet on the current test. When an item needs a genuinely unfamiliar or complex formula, that formula is supplied inside the item itself. Everything below is the kind of relationship worth knowing automatically, not a substitute for a reference sheet that doesn't exist on the actual test.
▸ Part I: Number & Quantity
| Relationship |
Where it was taught |
| i² = −1; multiply by the conjugate to divide complex numbers |
Real and Complex Numbers |
| When defined in the real numbers, x^(m/n) can be interpreted using an nth root and the power m; with an even root, the radicand must be nonnegative. Check candidate solutions after raising both sides of an equation to an even power |
Exponents and Radicals |
| Matrix multiplication requires columns of the first to equal rows of the second; determinant of [[a,b],[c,d]] = ad − bc |
Vectors and Matrices |
▸ Part II: Algebra
| Relationship |
Where it was taught |
| Multiplying or dividing an inequality by a negative reverses the inequality sign |
Linear Equations, Inequalities, and Systems |
| Discriminant b² − 4ac: positive gives two real solutions, zero gives one, negative gives two complex solutions; vertex at x = −b/2a |
Quadratics and the Discriminant |
| Factor by grouping; if x = a is a zero, (x − a) is a factor; rewrite exponential equations with a common base |
Polynomial, Radical, and Exponential Equations |
| |expression| = k splits into two cases; an absolute value can never equal a negative, so any case producing one is rejected |
Absolute Value |
▸ Part III: Functions
| Relationship |
Where it was taught |
| f(g(x)) means evaluate the inner function first; domain excludes zero denominators, negative values under an even root, and non-positive arguments of a logarithm |
Notation, Evaluation, and Composition |
| Roots, zeros, and x-intercepts name the same feature; end behavior is controlled by degree and leading-coefficient sign |
Linear, Quadratic, and Polynomial Functions |
| After common factors are canceled, a zero that remains in the denominator gives a vertical asymptote; a canceled denominator factor gives a hole. For horizontal asymptotes: numerator degree smaller gives y=0; equal degrees gives the ratio of leading coefficients; numerator degree larger gives no horizontal asymptote |
Rational, Radical, Exponential, and Logarithmic Functions |
| f(x+h) shifts horizontally, f(x)+k shifts vertically, −f(x) reflects over the x-axis, f(−x) reflects over the y-axis |
Transformations and Piecewise Functions |
| An inverse is found by swapping x and y and solving for the new y; verify with f(g(x)) = x and g(f(x)) = x |
Inverse Functions |
| Arithmetic series: Sₙ = (n/2)(a₁+aₙ); geometric series: Sₙ = a₁(1−rⁿ)/(1−r) for r≠1; Σ substitutes each index value in turn and adds the results |
Sequences and Series |
| For b>0 and b≠1, log_b(x)=k is equivalent to x=bᵏ. In the real-number setting, logarithm arguments must be positive. Where all logarithms are defined: log(ab)=log a+log b; log(a/b)=log a−log b; log(aⁿ)=n log a |
Logarithm Properties and Equations |
▸ Part IV: Geometry
| Relationship |
Where it was taught |
| Interior angles of a triangle sum to 180°; an exterior angle equals the sum of the two remote interior angles |
Lines, Angles, and Triangles |
| Similar figures have proportional sides and equal corresponding angles; scale factor applies directly to length and perimeter, and its square to area |
Congruence and Similarity |
| sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent; a²+b²=c² |
Right Triangles, Trigonometry, and the Unit Circle |
| (x−h)²+(y−k)²=r² for a circle; arc length = radius × central angle in radians |
Circles and Conics |
| Area and volume of composite figures are found by adding or subtracting the areas or volumes of their pieces |
Area, Volume, and Composite Figures |
| Distance = √[(x₂−x₁)²+(y₂−y₁)²]; midpoint = [(x₁+x₂)/2, (y₁+y₂)/2]. Parallel nonvertical lines have equal slopes. Perpendicular nonvertical, nonhorizontal lines have negative-reciprocal slopes; horizontal and vertical lines are the special perpendicular pair |
Coordinate Geometry |
| sin²θ+cos²θ=1; Law of Sines: a/sin A=b/sin B=c/sin C; Law of Cosines: c²=a²+b²−2ab cos C |
Trigonometric Identities and General Triangles |
| Rotation, reflection, and translation preserve size and shape; dilation by scale factor k multiplies coordinates by k, and multiplies area by k² |
Geometric Transformations |
| An inscribed angle is half the central angle intercepting the same arc; intersecting chords and secant/tangent segments follow power-of-a-point relationships |
Circle Theorems |
| A conditional statement's contrapositive shares its truth value; a counterexample disproves a statement by meeting its hypothesis but not its conclusion |
Proof, Reasoning, and Constructions |
▸ Part V: Statistics & Probability
| Relationship |
Where it was taught |
| Mean is sensitive to outliers; median usually is not; IQR = Q3 − Q1; right skew typically pulls the mean above the median, left skew below it |
Center and Spread |
| A randomized experiment supports a causal conclusion; an observational study supports only an association |
Data Collection and Study Design |
| Residual = actual − predicted; a strong correlation does not by itself establish causation; random residual scatter suggests a model fits well, a patterned residual suggests it does not |
Bivariate Data and Models |
| P(A or B) = P(A)+P(B)−P(A and B); P(A and B) = P(A)·P(B|A), which reduces to P(A)·P(B) when A and B are independent |
Probability and Sample Spaces |
| z = (x−μ)/σ; about 68%/95%/99.7% of values fall within 1/2/3 standard deviations of the mean in a roughly normal distribution |
Normal Distributions and Z-Scores |
| Plausible interval = estimate ± margin of error; a larger random sample lowers sampling variability, but sample size alone does not fix bias or guarantee representativeness; a claim is rejected when the observed result is unusually rare under simulations built from that claim, and otherwise the decision is to fail to reject it |
Introduction to Statistical Inference |
▸ Part VI: Integrating Essential Skills
| Relationship |
Where it was taught |
| Percent change is always measured against the original value, never the new one |
Rates, Percentages, and Proportional Reasoning |
| An area conversion factor must be squared and a volume conversion factor must be cubed, not applied as a single multiplier |
Measurement and Unit Conversion |
| Average speed is total distance divided by total time, not the simple average of two speeds |
Averages and Medians in Context |
None of the tables above replaces the teaching in their original chapters. If a relationship here doesn't immediately make sense, that's a signal to revisit the chapter it came from, not to memorize the row in isolation.
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