📚 SAT Math Essential Formulas and Concepts Summary | SAT数学必备公式与概念汇总
The SAT Math section tests a wide range of mathematical concepts, from basic algebra to advanced trigonometry. A solid grasp of essential formulas and concepts is crucial for efficiency and accuracy. This guide consolidates the must‑know formulas and key ideas, organized by topic, to help you confidently tackle any problem. Use it as a quick reference while practicing and before the test to reinforce your memory.
SAT数学部分考查从基础代数到高级三角学的广泛概念。熟练掌握必备公式和概念对于提高解题效率和准确性至关重要。本指南按主题汇总了必知的公式与核心思路，助你自信应对各类题目。可在练习中及考前快速查阅，以强化记忆。
1. Linear Equations & Inequalities | 线性方程与不等式
Linear equations appear in various forms. The slope‑intercept form y = mx + b identifies the slope m and y‑intercept b. A line’s slope is constant and can be found from two points (x₁, y₁) and (x₂, y₂).
线性方程有多种形式。斜截式 y = mx + b 中，m 为斜率，b 为y轴截距。直线的斜率恒定，可由两点 (x₁, y₁) 与 (x₂, y₂) 求出。
Slope: m = (y₂ – y₁) / (x₂ – x₁)
斜率：m = (y₂ – y₁) / (x₂ – x₁)
Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (m₁ × m₂ = -1). A horizontal line has zero slope, a vertical line has undefined slope.
平行线斜率相等；垂直线的斜率互为负倒数（m₁ × m₂ = -1）。水平线斜率为零，竖直线斜率无定义。
Solving linear inequalities follows the same steps as equations, except that multiplying or dividing by a negative number reverses the inequality sign.
解线性不等式步骤与方程相同，但乘以或除以负数时，不等号方向反转。
2. Systems of Equations | 方程组
A system of linear equations can be solved by substitution, elimination, or graphing. The solution is the intersection point. A system has one solution, no solution (parallel lines), or infinitely many solutions (coincident lines).
线性方程组可通过代入法、消元法或图像法求解，解为交点。方程组可能有一个解（相交）、无解（平行）或无穷多解（重合直线）。
For word problems, define variables clearly. The SAT often asks about the number of solutions by comparing ratios of coefficients. For a system a₁x + b₁y = c₁ and a₂x + b₂y = c₂, consistent systems depend on a₁/a₂ and b₁/b₂ relative to c₁/c₂.
文字题需清晰设定变量。SAT常通过比较系数比来考查解的个数。对于方程组 a₁x + b₁y = c₁ 与 a₂x + b₂y = c₂，由 a₁/a₂、b₁/b₂ 和 c₁/c₂ 的关系判断解的情况。
3. Quadratic Equations & Functions | 二次方程与函数
The standard form is ax² + bx + c = 0. The quadratic formula provides solutions even when factoring fails.
标准形式为 ax² + bx + c = 0。当因式分解困难时，求根公式总能给出解。
x = (-b ± √(b² – 4ac)) / (2a)
求根公式：x = (-b ± √(b² – 4ac)) / (2a)
The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 → two distinct real roots; Δ = 0 → one real root (double root); Δ < 0 → two complex conjugate roots. The sum of roots is -b/a and the product is c/a.
判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个实根（二重根）；Δ < 0 有两个共轭复根。根之和为 -b/a，根之积为 c/a。
A quadratic function can be written in vertex form: f(x) = a(x – h)² + k, where (h, k) is the vertex. The axis of symmetry is x = h. The vertex can also be found using h = -b/(2a).
二次函数可写成顶点式：f(x) = a(x – h)² + k，其中 (h, k) 为顶点。对称轴为 x = h。顶点坐标也可由 h = -b/(2a) 求出。
4. Exponents, Radicals & Polynomials | 指数、根式与多项式
Master exponent rules: xᵃ · xᵇ = xᵃ⁺ᵇ, (xᵃ)ᵇ = xᵃᵇ, xᵃ / xᵇ = xᵃ⁻ᵇ (x ≠ 0), x⁰ = 1, x⁻ⁿ = 1/xⁿ. Fractional exponents connect to radicals: x^(1/n) = ⁿ√x, and x^(m/n) = ⁿ√(xᵐ).
指数法则：xᵃ · xᵇ = xᵃ⁺ᵇ, (xᵃ)ᵇ = xᵃᵇ, xᵃ / xᵇ = xᵃ⁻ᵇ (x ≠ 0), x⁰ = 1, x⁻ⁿ = 1/xⁿ。分数指数与根式关联：x^(1/n) = ⁿ√x, x^(m/n) = ⁿ√(xᵐ)。
When adding or subtracting polynomials, combine like terms. Watch out for standard SAT pitfalls such as (x + y)² ≠ x² + y². Instead, (x + y)² = x² + 2xy + y².
多项式加减时合并同类项。注意SAT常见陷阱，例如 (x + y)² ≠ x² + y²，正确应为 (x + y)² = x² + 2xy + y²。
Factoring is essential: difference of squares a² – b² = (a – b)(a + b); perfect square trinomials a² ± 2ab + b² = (a ± b)². For higher‑degree polynomials, look for greatest common factors and grouping.
因式分解至关重要：平方差 a² – b² = (a – b)(a + b)；完全平方式 a² ± 2ab + b² = (a ± b)²。对高次多项式，先提取公因式，再用分组分解。
5. Rational Expressions & Equations | 有理表达式与方程
A rational expression is a fraction of polynomials. Simplify by factoring and cancelling common factors, but note the domain restrictions (denominator ≠ 0). Rational equations are solved by multiplying both sides by the least common denominator and checking for extraneous solutions.
有理表达式为多项式分式。通过因式分解并约去公因子来化简，但需注明定义域限制（分母 ≠ 0）。解有理方程时，两边同乘最简公分母，并检验增根。
Operations: a/b + c/d = (ad + bc) / bd. Division by a rational expression is multiplication by its reciprocal. Always simplify to lowest terms.
运算：a/b + c/d = (ad + bc) / bd。除以有理表达式等于乘以其倒数。结果必须化为最简形式。
6. Complex Numbers | 复数
Complex numbers are of the form a + bi, where i² = -1. When simplifying expressions with i, use i² = -1, i³ = -i, i⁴ = 1. The conjugate of a + bi is a – bi, and (a + bi)(a – bi) = a² + b².
复数形如 a + bi，其中 i² = -1。化简含 i 的表达式时，使用 i² = -1, i³ = -i, i⁴ = 1。a + bi 的共轭为 a – bi，且 (a + bi)(a – bi) = a² + b²。
To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator. On the SAT, complex numbers often appear in solving quadratics or simplifying powers of i.
复数相除时，将分子分母同乘分母的共轭。SAT中复数常出现在解二次方程或化简 i 的幂次中。
7. Functions & Graphs | 函数与图像
A function relates each input to exactly one output. Function notation f(x) is common. The domain is the set of all possible x‑values; the range is the set of all possible f(x) values. Be aware of domain restrictions such as division by zero and even roots of negative numbers.
函数将每个输入对应到唯一输出。常用函数记号 f(x)。定义域为所有可能的 x 值集合；值域为所有可能的 f(x) 值集合。注意定义域限制，如分母不为零、偶次根下非负。
Graph transformations: f(x) + c shifts up, f(x) – c shifts down; f(x + c) shifts left, f(x – c) shifts right; -f(x) reflects across the x‑axis; f(-x) reflects across the y‑axis; a·f(x) stretches vertically if |a| > 1, compresses if 0 < |a| < 1.
图像变换：f(x) + c 上移，f(x) – c 下移；f(x + c) 左移，f(x – c) 右移；-f(x) 关于 x 轴反射；f(-x) 关于 y 轴反射；a·f(x) 当 |a|>1 时垂直拉伸，0<|a|<1 时垂直压缩。
Composite functions f(g(x)) apply g first, then f. Inverse functions reverse the mapping; graphically, the inverse is a reflection across the line y = x.
复合函数 f(g(x)) 先应用 g，再应用 f。反函数反转映射关系；图像上，反函数关于直线 y = x 对称。
8. Geometry: Lines, Angles & Triangles | 几何：线、角与三角形
Key angle facts: angles on a straight line sum to 180°; vertically opposite angles are equal. When parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary (sum to 180°).
重要角度性质：直线上的角之和为180°；对顶角相等。平行线被截线所截时，同位角相等，内错角相等，同旁内角互补（和为180°）。
Triangle sum: interior angles add to 180°. In a right triangle, Pythagoras’ theorem applies: a² + b² = c², where c is the hypotenuse. Special right triangles: 30°‑60°‑90° triangle has side ratio 1 : √3 : 2; 45°‑45°‑90° triangle has ratio 1 : 1 : √2.
三角形内角和为180°。直角三角形中，勾股定理成立：a² + b² = c²，c 为斜边。特殊直角三角形：30°‑60°‑90° 三角形三边比例为 1 : √3 : 2；45°‑45°‑90° 三角形比例为 1 : 1 : √2。
Pythagorean Theorem: a² + b² = c²
勾股定理：a² + b² = c²
Similar triangles have congruent angles and proportional sides. The area of a triangle is (1/2) × base × height. The perimeter of any shape is the sum of all side lengths.
相似三角形对应角相等，对应边成比例。三角形面积为 (1/2) × 底 × 高。任意图形的周长为所有边长之和。
9. Circles & Volume | 圆与体积
Circle formulas: Circumference C = 2πr or πd; area A = πr². An arc length is a fraction of the circumference: s = (θ/360°) · 2πr when θ is in degrees, or s = rθ when θ is in radians. Sector area is similarly proportional.
圆公式：周长 C = 2πr 或 πd；面积 A = πr²。弧长是周长的一部分：当 θ 为度数时，s = (θ/360°) · 2πr；当 θ 为弧度时，s = rθ。扇形面积同样按比例计算。
Arc length (radians): s = rθ
弧长（弧度制）：s = rθ
A tangent to a circle is perpendicular to the radius at the point of tangency. The equation of a circle with center (h, k) and radius r is (x – h)² + (y – k)² = r².
圆的切线垂直于经过切点的半径。圆心为 (h, k)、半径为 r 的圆方程为 (x – h)² + (y – k)² = r²。
Volume of a rectangular prism: V = lwh; cylinder: V = πr²h; cone: V = (1/3)πr²h; sphere: V = (4/3)πr³. The SAT provides these reference formulas, but knowing them saves time.
体积：长方体 V = lwh；圆柱 V = πr²h；圆锥 V = (1/3)πr²h；球体 V = (4/3)πr³。SAT提供这些参考公式，但熟记可节省时间。
10. Coordinate Geometry | 坐标几何
The distance between two points (x₁, y₁) and (x₂, y₂) is d = √[(x₂ – x₁)² + (y₂ – y₁)²]. The midpoint is M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点距离：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。中点坐标：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
To find the intersection point of two lines, solve the system of equations. For a perpendicular bisector of a segment, combine the midpoint formula with the negative reciprocal slope.
求两直线交点，解方程组即可。线段的垂直平分线需结合中点公式与负倒数斜率。
X‑intercept occurs when y = 0; y‑intercept when x = 0. Rearranging equations to slope‑intercept form helps sketch graphs.
x轴截距由 y = 0 求出；y轴截距由 x = 0 求出。将方程化为斜截式有助于画出图像。
11. Trigonometry & Radians | 三角学与弧度
Basic right‑triangle definitions: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The mnemonic SOHCAHTOA is helpful.
基本直角三角形定义：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。可用口诀SOHCAHTOA辅助记忆。
Complementary angle identities: sin(90° – θ) = cos θ, cos(90° – θ) = sin θ. In any triangle, the Law of Sines and Law of Cosines may appear, but are less frequent on the SAT.
余角恒等式：sin(90° – θ) = cos θ, cos(90° – θ) = sin θ。任意三角形可能出现正弦定理和余弦定理，但在SAT中较少见。
Radian measure: 180° = π radians. To convert, multiply by π/180 or 180/π. The unit circle gives precise trig values at key angles (π/6, π/4, π/3, etc.).
弧度制：180° = π 弧度。转换时乘以 π/180 或 180/π。单位圆给出关键角（π/6、π/4、π/3等）的准确三角函数值。
Fundamental identity: sin²θ + cos²θ = 1. Also, tan θ = sin θ / cos θ. These are useful for simplifying expressions.
基本恒等式：sin²θ + cos²θ = 1。另有 tan θ = sin θ / cos θ。这些关系用于化简表达式。
12. Data Analysis & Statistics | 数据分析与统计
Measures of center: mean (average) = sum/n, median = middle value when sorted, mode = most frequent value. The SAT often tests how an outlier affects mean versus median.
集中量数：平均数 = 总和/n，中位数 = 排序后的中间值，众数 = 出现频率最高的值。SAT常考查异常值对平均数和中位数的影响。
Measures of spread: range = maximum – minimum; standard deviation measures how spread out data are around the mean. A low standard deviation means data points are clustered, a high one means more variation.
离散量数：极差 = 最大值 – 最小值；标准差衡量数据围绕平均数的分散程度。标准差小表示数据聚集，标准差大表示变异性大。
Probability basics: P(event) = favorable outcomes / total outcomes. For independent events, P(A and B) = P(A) × P(B). For two‑way tables, marginal, joint, and conditional probabilities appear.
概率基础：P(事件) = 有利结果数 / 总结果数。独立事件中，P(A 且 B) = P(A) × P(B)。双向表中常考查边缘概率、联合概率和条件概率。
Scatterplots and lines of best fit: slope indicates the change in y per unit change in x. Correlation does not imply causation. Understanding how to interpret survey sample results and margin of error is also tested.
散点图与最佳拟合线：斜率表示 x 每增加一单位 y 的变动量。相关关系不代表因果关系。还需理解如何解读调查抽样结果和误差范围。
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