SAT2 Math Core Formulas Review | SAT2数学核心公式梳理

📚 SAT2 Math Core Formulas Review | SAT2数学核心公式梳理

This article provides a concise yet comprehensive review of the essential formulas for the SAT Subject Test in Mathematics Level 2. All formulas are organized by topic to help you revise efficiently and avoid common pitfalls on exam day.

本文为SAT2数学（Level 2）考生提供一份精炼而全面的核心公式汇总。所有公式按专题分类整理，帮助你高效复习，避免考试当天出现常见失误。

1. Algebra & Factoring | 代数与因式分解

Quadratic formula: For ax² + bx + c = 0, the roots are given by:

x = (−b ± √(b² − 4ac)) / 2a

The discriminant Δ = b² − 4ac determines the nature of roots: if Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, two complex conjugate roots.

二次公式：对于 ax² + bx + c = 0，求根公式为：

x = (−b ± √(b² − 4ac)) / 2a

判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 时有两个不等实根；Δ = 0 时有一个重根；Δ < 0 时有两个共轭复根。

Factoring identities:

a² − b² = (a − b)(a + b)
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²

因式分解恒等式：

a² − b² = (a − b)(a + b)
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²

2. Functions & Transformations | 函数与变换

Absolute value: |x| = x if x ≥ 0, and |x| = −x if x < 0. The graph is V-shaped with vertex at (0, 0).

绝对值：|x| = x（当 x ≥ 0），|x| = −x（当 x < 0）。其图像为 V 形，顶点在 (0, 0)。

Composition and inverse: (f ∘ g)(x) = f(g(x)). If f has an inverse, then f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. The domain of f⁻¹ equals the range of f, and vice versa.

复合与反函数：(f ∘ g)(x) = f(g(x))。若 f 存在反函数，则 f⁻¹(f(x)) = x，且 f(f⁻¹(x)) = x。f⁻¹ 的定义域等于 f 的值域，反之亦然。

Transformations: Given y = f(x):

y = f(x) + k shifts upward by k units (k > 0)
y = f(x) − k shifts downward by k units
y = f(x − h) shifts right by h units
y = f(x + h) shifts left by h units
y = −f(x) reflects across the x-axis
y = f(−x) reflects across the y-axis
y = a·f(x) stretches vertically if |a| > 1, compresses if 0 < |a| < 1

图像变换：已知 y = f(x)：

y = f(x) + k 向上平移 k 个单位（k > 0）
y = f(x) − k 向下平移 k 个单位
y = f(x − h) 向右平移 h 个单位
y = f(x + h) 向左平移 h 个单位
y = −f(x) 关于 x 轴对称翻折
y = f(−x) 关于 y 轴对称翻折
y = a·f(x) 当 |a| > 1 时纵向拉伸；当 0 < |a| < 1 时纵向压缩

Exponential and logarithmic: y = aˣ has domain (−∞, ∞), range (0, ∞), and horizontal asymptote y = 0. The inverse is y = logₐx, with domain (0, ∞) and range (−∞, ∞). Key rules:

指数与对数：y = aˣ 的定义域为 (−∞, ∞)，值域为 (0, ∞)，水平渐近线为 y = 0。其反函数为 y = logₐx，定义域为 (0, ∞)，值域为 (−∞, ∞)。关键运算法则：

logₐ(MN) = logₐM + logₐN
logₐ(M/N) = logₐM − logₐN
logₐ(Mⁿ) = n·logₐM
Change of base: logₐb = log b / log a

logₐ(MN) = logₐM + logₐN
logₐ(M/N) = logₐM − logₐN
logₐ(Mⁿ) = n·logₐM
换底公式：logₐb = log b / log a

3. Coordinate Geometry | 坐标几何

Distance and midpoint: The distance between A(x₁, y₁) and B(x₂, y₂) is:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

距离与中点：A(x₁, y₁) 与 B(x₂, y₂) 之间的距离为：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。

Lines: Slope m = (y₂ − y₁)/(x₂ − x₁). Equation forms: point-slope y − y₁ = m(x − x₁); slope-intercept y = mx + b. Two lines are parallel if m₁ = m₂; perpendicular if m₁·m₂ = −1.

直线：斜率 m = (y₂ − y₁)/(x₂ − x₁)。方程形式：点斜式 y − y₁ = m(x − x₁)；斜截式 y = mx + b。两直线平行当且仅当 m₁ = m₂；垂直当且仅当 m₁·m₂ = −1。

Circles: Center (h, k), radius r:

(x − h)² + (y − k)² = r²

圆：圆心 (h, k)，半径 r：

(x − h)² + (y − k)² = r²

Parabola: Vertex form y = a(x − h)² + k. The vertex is (h, k); axis of symmetry is x = h; if a > 0 it opens upward, if a < 0 it opens downward.

抛物线：顶点式 y = a(x − h)² + k。顶点为 (h, k)，对称轴为 x = h；若 a > 0 开口向上，若 a < 0 开口向下。

Ellipse: Standard form (x − h)²/a² + (y − k)²/b² = 1 (a > b). Foci lie on the major axis, with c² = a² − b².

椭圆：标准方程 (x − h)²/a² + (y − k)²/b² = 1（a > b）。焦点位于长轴上，且 c² = a² − b²。

Hyperbola: Standard form (x − h)²/a² − (y − k)²/b² = 1. Asymptotes are y − k = ±(b/a)(x − h). Here c² = a² + b².

双曲线：标准方程 (x − h)²/a² − (y − k)²/b² = 1。渐近线为 y − k = ±(b/a)(x − h)。这里 c² = a² + b²。

4. Geometry | 平面几何

Triangles: Area = ½ × base × height. Heron’s formula: Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2. Pythagorean theorem: a² + b² = c² for right triangles. Special right triangles: 30°‑60°‑90° sides are x, x√3, 2x; 45°‑45°‑90° sides are x, x, x√2.

三角形：面积 = ½ × 底 × 高。海伦公式：面积 = √[s(s − a)(s − b)(s − c)]，其中 s = (a + b + c)/2。勾股定理：直角三角形中 a² + b² = c²。特殊直角三角形：30°‑60°‑90° 的三边比为 x : x√3 : 2x；45°‑45°‑90° 的三边比为 x : x : x√2。

Quadrilaterals: Rectangle area = lw; parallelogram area = bh; trapezoid area = ½(b₁ + b₂)h; rhombus area = ½d₁d₂ (diagonals product).

四边形：矩形面积 = lw；平行四边形面积 = bh；梯形面积 = ½(b₁ + b₂)h；菱形面积 = ½d₁d₂（两对角线乘积的一半）。

Circles: Circumference C = 2πr; area A = πr²; arc length L = rθ (θ in radians); sector area = ½r²θ.

圆：周长 C = 2πr；面积 A = πr²；弧长 L = rθ（θ 以弧度制表示）；扇形面积 = ½r²θ。

Solids: Volume of rectangular prism = lwh; cylinder V = πr²h; cone V = ⅓πr²h; sphere V = ⁴⁄₃πr³. Surface areas: sphere S = 4πr²; cylinder lateral S = 2πrh; cone lateral S = πrl (l = slant height).

立体几何：长方体体积 = lwh；圆柱体积 V = πr²h；圆锥体积 V = ⅓πr²h；球体积 V = ⁴⁄₃πr³。表面积：球 S = 4πr²；圆柱侧面积 S = 2πrh；圆锥侧面积 S = πrl（l 为母线长）。

5. Trigonometry | 三角学

Basic ratios: sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent. Reciprocal identities: csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ.

基本比例：sin θ = 对边/斜边；cos θ = 邻边/斜边；tan θ = 对边/邻边。倒数恒等式：csc θ = 1/sin θ；sec θ = 1/cos θ；cot θ = 1/tan θ。

Pythagorean identity: sin²θ + cos²θ = 1. Also: 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.

毕达哥拉斯恒等式：sin²θ + cos²θ = 1。此外：1 + tan²θ = sec²θ；1 + cot²θ = csc²θ。

Angle addition:

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

角的加法公式：

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

Double angles: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

二倍角公式：sin 2θ = 2 sin θ cos θ；cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。

Law of Sines and Cosines: In triangle ABC with sides a, b, c opposite angles A, B, C respectively:

a/sin A = b/sin B = c/sin C

c² = a² + b² − 2ab·cos C

正弦定理与余弦定理：在三角形 ABC 中，边 a, b, c 分别对角 A, B, C：

a/sin A = b/sin B = c/sin C

c² = a² + b² − 2ab·cos C

Radian conversion: To convert degrees to radians, multiply by π/180°; to convert radians to degrees, multiply by 180°/π. Key values: 180° = π radians; 360° = 2π radians.

弧度换算：将角度乘以 π/180° 可化为弧度；将弧度乘以 180°/π 可化为角度。关键值：180° = π 弧度；360° = 2π 弧度。

6. Sequences & Series | 数列与级数

Arithmetic sequence: aₙ = a₁ + (n − 1)d; sum of first n terms: Sₙ = n/2 · [2a₁ + (n − 1)d] = n/2 · (a₁ + aₙ).

等差数列：aₙ = a₁ + (n − 1)d；前 n 项和：Sₙ = n/2 · [2a₁ + (n − 1)d] = n/2 · (a₁ + aₙ)。

Geometric sequence: aₙ = a₁·rⁿ⁻¹; sum of first n terms: Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. Infinite geometric series converges to S = a₁/(1 − r) when |r| < 1.

等比数列：aₙ = a₁·rⁿ⁻¹；前 n 项和：Sₙ = a₁(1 − rⁿ)/(1 − r)（r ≠ 1）。当 |r| < 1 时，无穷等比级数收敛于 S = a₁/(1 − r)。

Factorials and binomial coefficients: n! = n × (n − 1) × … × 1. The binomial coefficient C(n, k) = n! / [k!(n − k)!]. The binomial theorem:

(a + b)ⁿ = Σ C(n, k)·aⁿ⁻ᵏ·bᵏ

阶乘与组合数：n! = n × (n − 1) × … × 1。组合数 C(n, k) = n! / [k!(n − k)!]。二项式定理：

(a + b)ⁿ = Σ C(n, k)·aⁿ⁻ᵏ·bᵏ

7. Statistics & Probability | 统计与概率

Mean, median, mode: The mean is the sum of data divided by the number of data points. The median is the middle value when data is ordered; for an even number of values, take the average of the two middle values. The mode is the most frequently occurring value.

均值、中位数、众数：均值是数据之和除以数据个数。中位数是将数据按大小排列后的中间值；若数据个数为偶数，则取中间两个数的平均值。众数是出现频率最高的数值。

Variance and standard deviation: For a population with mean μ and n data points:

σ² = [Σ(xᵢ − μ)²] / n

Standard deviation σ = √σ². Sample variance uses denominator n − 1.

方差与标准差：对均值为 μ、含 n 个数据点的总体：

σ² = [Σ(xᵢ − μ)²] / n

标准差 σ = √σ²。样本方差的分母为 n − 1。

Probability rules: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0. If A and B are independent, P(A ∩ B) = P(A)·P(B). Conditional probability: P(A|B) = P(A ∩ B)/P(B).

概率法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若 A 与 B 互斥，则 P(A ∩ B) = 0。若 A 与 B 独立，则 P(A ∩ B) = P(A)·P(B)。条件概率：P(A|B) = P(A ∩ B)/P(B)。

Counting: Permutations of n distinct objects taken r at a time: P(n, r) = n!/(n − r)!. Combinations: C(n, r) = n!/[r!(n − r)!].

计数原理：从 n 个不同对象中取出 r 个的排列数：P(n, r) = n!/(n − r)!。组合数：C(n, r) = n!/[r!(n − r)!]。

Normal distribution: For a standard normal distribution, the z-score is z = (x − μ)/σ. Approximately 68% of data lies within 1 standard deviation, 95% within 2, and 99.7% within 3.

正态分布：标准正态分布的 z 分数为 z = (x − μ)/σ。约 68% 的数据落在 1 个标准差内，95% 落在 2 个标准差内，99.7% 落在 3 个标准差内。

8. Complex Numbers & Vectors | 复数与向量

Complex numbers: A complex number z = a + bi, where i² = −1. The modulus is |z| = √(a² + b²). The conjugate is z̄ = a − bi. For multiplication, (a + bi)(c + di) = (ac − bd) + (ad + bc)i.

复数：复数 z = a + bi，其中 i² = −1。模为 |z| = √(a² + b²)。共轭复数为 z̄ = a − bi。乘法运算：(a + bi)(c + di) = (ac − bd) + (ad + bc)i。

Polar form: z = r(cos θ + i sin θ) = r·cis θ, where r = |z| and θ = arg(z). De Moivre’s theorem: zⁿ = rⁿ(cos nθ + i sin nθ).

极坐标形式：z = r(cos θ + i sin θ) = r·cis θ，其中 r = |z|，θ = arg(z)。棣莫弗定理：zⁿ = rⁿ(cos nθ + i sin nθ)。

Vectors: For vectors u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the dot product is u·v = u₁v₁ + u₂v₂ = |u||v|cos θ, where θ is the angle between them. The magnitude is |u| = √(u₁² + u₂²).

向量：对于向量 u = ⟨u₁, u₂⟩ 和 v = ⟨v₁, v₂⟩，点积为 u·v = u₁v₁ + u₂v₂ = |u||v|cos θ，其中 θ 为两向量夹角。向量模为 |u| = √(u₁² + u₂²)。

Parametric equations: To convert a parametric curve x = f(t), y = g(t) to Cartesian form, eliminate the parameter t. The slope of the tangent is dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

参数方程：将参数曲线 x = f(t)，y = g(t) 化为直角坐标方程时，消去参数 t 即可。切线斜率为 dy/dx = (dy/dt)/(dx/dt)，前提是 dx/dt ≠ 0。

9. Limits & Introduction to Calculus | 极限与微积分入门

Limits: The limit of a function f(x) as x approaches a is L if the values of f(x) get arbitrarily close to L for x sufficiently close to a (from both sides). Key rules:

极限：当 x 趋于 a 时，若 f(x) 的值无限接近 L，则称极限为 L（需从两侧接近）。关键法则：

lim [f(x) ± g(x)] = lim f(x) ± lim g(x)
lim [c·f(x)] = c·lim f(x)
lim [f(x)·g(x)] = lim f(x) · lim g(x)
lim [f(x)/g(x)] = lim f(x) / lim g(x) (if lim g(x) ≠ 0)
Special limit: lim (sin x)/x = 1 as x → 0

lim [f(x) ± g(x)] = lim f(x) ± lim g(x)
lim [c·f(x)] = c·lim f(x)
lim [f(x)·g(x)] = lim f(x) · lim g(x)
lim [f(x)/g(x)] = lim f(x) / lim g(x)（若 lim g(x) ≠ 0）
特殊极限：当 x → 0 时，lim (sin x)/x = 1

Derivatives: The derivative f'(x) represents the instantaneous rate of change, or the slope of the tangent line. Basic rules:

导数：导数 f'(x) 表示瞬时变化率，即切线的斜率。基本求导法则：

d/dx [xⁿ] = n·xⁿ⁻¹
d/dx [sin x] = cos x; d/dx [cos x] = −sin x
d/dx [eˣ] = eˣ; d/dx [ln x] = 1/x
Product rule: (fg)’ = f’g + fg’
Quotient rule: (f/g)’ = (f’g − fg’)/g²
Chain rule: (f(g(x)))’ = f'(g(x))·g'(x)

d/dx [xⁿ] = n·xⁿ⁻¹
d/dx [sin x] = cos x；d/dx [cos x] = −sin x
d/dx [eˣ] = eˣ；d/dx [ln x] = 1/x
乘积法则：(fg)’ = f’g + fg’
商法则：(f/g)’ = (f’g − fg’)/g²
链式法则：(f(g(x)))’ = f'(g(x))·g'(x)

10. Final Exam Tips | 考试最终建议

Know your calculator: SAT2 Math Level 2 heavily relies on a graphing calculator. Practice entering statistical calculations, matrix operations, and solving equations efficiently. Always verify that the calculator is in the correct mode (degrees or radians).

熟悉计算器：SAT2数学 Level 2 考试非常依赖图形计算器。务必练习统计计算、矩阵运算和高效求解方程。始终确认计算器处于正确的角度模式（角度制或弧度制）。

Memorize the formula sheet: While the exam provides some formulas, many critical identities such as double-angle formulas, conic section equations, and normal distribution percentages must be memorized. Write these down repeatedly before exam day.

牢记公式表：虽然考试会提供部分公式，但许多关键恒等式如二倍角公式、圆锥曲线方程、正态分布百分比都需要记忆。考试前请反复默写这些公式。

Time management: The test has 50 questions in 60 minutes. Answer the easy questions first, skip and mark difficult ones, then return if time permits. There is a penalty for wrong answers, so guess only when you can eliminate at least one option confidently.

时间管理：考试共 50 题，时长 60 分钟。先做简单题，跳过并标记难题，如有剩余时间再回头作答。答错会扣分，因此只有在能排除至少一个选项时才值得猜题。

Common pitfalls: Watch out for sign errors when applying the quadratic formula; remember that logₐx is undefined for x ≤ 0; ensure your calculator is in radian mode for trig problems unless the problem specifies degrees; and always double-check the domain of a function before determining its range.

常见陷阱：应用二次公式时注意符号错误；注意 logₐx 在 x ≤ 0 时无定义；除非题目注明使用角度制，否则三角题务必确保计算器处于弧度制；在确定值域之前，务必先检查函数的定义域。

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SAT2 Math Core Formulas Review | SAT2数学核心公式梳理