📚 SAT Math Formula Master Guide | SAT数学公式盘点：必背核心公式与解题应用
The SAT Math section tests your ability to apply formulas quickly and accurately under timed conditions. This guide compiles every essential formula you must memorize, organized by topic, with practical tips for applying them to real test questions.
SAT数学部分考查你在限时条件下快速、准确运用公式的能力。本指南按主题整理了所有必须牢记的核心公式，并结合真题题型提供实用解题技巧，助你在考场上从容应对。
1. Linear Equations & Slope | 线性方程与斜率
Linear equations are the backbone of the Heart of Algebra questions. The slope-intercept form and point-slope form are the two most frequently used equations on the exam.
线性方程是代数核心（Heart of Algebra）题型的基石。斜截式和点斜式是考试中最常使用的两种方程形式。
- Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
- 斜截式：y = mx + b，其中 m 为斜率，b 为 y 轴截距。
- Point-slope form: y − y₁ = m(x − x₁), where (x₁, y₁) is a point on the line.
- 点斜式：y − y₁ = m(x − x₁)，其中 (x₁, y₁) 为直线上一点。
- Slope formula: m = (y₂ − y₁) / (x₂ − x₁) = Δy / Δx.
- 斜率公式：m = (y₂ − y₁) / (x₂ − x₁) = Δy / Δx。
m = (y₂ − y₁) ÷ (x₂ − x₁)
× m₂ = −1). These relationships allow you to solve many questions without full equations.
两条直线平行时，斜率相等（m₁ = m₂）；垂直时，斜率互为负倒数（m₁ × m₂ = −1）。掌握这些关系，许多题目无需列出完整方程即可快速求解。
2. Systems of Equations | 方程组
Systems of equations questions ask you to find the point(s) where two equations intersect. Three methods appear repeatedly: substitution, elimination, and graphing.
方程组题目要求你找到两个方程的交点。三种方法反复出现：代入法、消元法和图像法。
- Substitution: Solve one equation for one variable, then substitute into the other.
- 代入法：先从一个方程中解出一个变量，再代入另一个方程。
- Elimination: Add or subtract equations to cancel one variable.
- 消元法：将两个方程相加或相减，消去一个变量。
- Number of solutions: One solution (intersecting lines), zero solutions (parallel lines), or infinite solutions (identical lines).
- 解的个数：一个解（两线相交）、零个解（两线平行）或无穷多个解（两线重合）。
For the no-calculator section, elimination is usually faster. For the calculator section, graphing both equations and finding the intersection point is often the most reliable strategy.
在不可使用计算器的部分，消元法通常更快；在可使用计算器的部分，画出两条方程的图像并找到交点往往是最可靠的策略。
3. Quadratic Equations | 二次方程
Quadratics appear in both the no-calculator and calculator sections. You must know the standard form, vertex form, and the quadratic formula.
二次方程在不可用计算器和可用计算器两部分都会出现。你必须掌握标准形式、顶点形式以及求根公式。
Standard form: ax² + bx + c = 0
Quadratic formula: x = [−b ± √b² − 4ac] ÷ 2a
求根公式：x = [−b ± √b² − 4ac] ÷ 2a
- Discriminant: D = b² − 4ac. If D > 0, two real roots; if D = 0, one real root; if D < 0, no real roots.
- 判别式：D = b² − 4ac。若 D > 0，有两个实根；若 D = 0，有一个实根；若 D < 0，无实根。
- Vertex form: y = a(x − h)² + k, where (h, k) is the vertex.
- 顶点式：y = a(x − h)² + k，其中 (h, k) 为顶点坐标。
- Axis of symmetry: x = −b / (2a).
- 对称轴：x = −b / (2a)。
- Sum of roots: x₁ + x₂ = −b / a. Product of roots: x₁ × x₂ = c / a.
- 两根之和：x₁ + x₂ = −b / a。两根之积：x₁ × x₂ = c / a。
The vertex is the maximum or minimum point of the parabola. If a > 0, the parabola opens upward and the vertex is a minimum; if a < 0, it opens downward and the vertex is a maximum.
顶点是抛物线的最大值或最小值点。若 a > 0，抛物线开口向上，顶点为最小值；若 a < 0，开口向下，顶点为最大值。
4. Exponents & Radicals | 指数与根式
Exponent rules appear in nearly every SAT math section. Memorizing these properties is non-negotiable.
指数法则几乎出现在SAT数学的每一部分。牢记以下运算法则是必须的。
| Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ | 积的法则：aᵐ × aⁿ = aᵐ⁺ⁿ |
| Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 商的法则：aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power rule: (aᵐ)ⁿ = aᵐⁿ | 幂的法则：(aᵐ)ⁿ = aᵐⁿ |
| Zero exponent: a⁰ = 1 (a ≠ 0) | 零次幂：a⁰ = 1（a ≠ 0） |
| Negative exponent: a⁻ⁿ = 1 / aⁿ | 负指数：a⁻ⁿ = 1 / aⁿ |
| Fractional exponent: am/n = ⁿ√aᵐ | 分数指数：am/n = ⁿ√aᵐ |
A common SAT trap involves simplifying expressions with fractional exponents. For example, x3/2 means (√x)³, which equals √x³. Both interpretations are correct.
SAT常见的陷阱涉及分数指数化简。例如，x3/2 表示 (√x)³，也等于 √x³。两种理解都是正确的。
5. Exponential Growth & Decay | 指数增长与衰减
Real-world applications of exponentials appear frequently. The general formula models population growth, radioactive decay, and financial interest.
指数函数的实际应用频繁出现。通用公式可建模人口增长、放射性衰变和金融利息等问题。
y = A(1 ± r)ᵗ
- A = initial amount (初始量)
- r = rate of growth (+) or decay (−) (增长率 + 或衰减率 −)
- t = number of time periods (时间周期数)
- y = final amount (最终量)
If the rate is 5% growth per year, use (1 + 0.05)ᵗ. If the rate is 3% decay, use (1 − 0.03)ᵗ. The key is converting percentages to decimals before plugging in.
若年增长率为5%，使用 (1 + 0.05)ᵗ；若衰减率为3%，使用 (1 − 0.03)ᵗ。关键在于代入前将百分比转换为小数。
6. Statistics & Probability | 统计与概率
Statistics questions test your ability to interpret data, calculate averages, and understand probability in context.
统计题考查你解读数据、计算平均值和理解情境概率的能力。
- Mean: sum of all values ÷ number of values.
- 平均数：所有数值之和 ÷ 数值的个数。
- Median: the middle value when data is ordered from least to greatest.
- 中位数：数据从小到大排列后的中间值。
- Mode: the most frequently occurring value.
- 众数：出现频率最高的数值。
- Range: maximum value − minimum value.
- 极差：最大值 − 最小值。
- Probability: P(A) = number of favorable outcomes ÷ total possible outcomes.
- 概率：P(A) = 有利结果数 ÷ 所有可能结果数。
For probability questions involving "or," B) = P(A) + P(B) − P(A and B). This formula prevents double-counting when events overlap.
涉及"或"的概率题，记住：P(A或B) = P(A) + P(B) − P(A且B)。这个公式防止事件重叠时重复计数。
7. Geometry: Area & Perimeter | 几何：面积与周长
Geometry formulas are provided in the SAT reference box, but knowing them cold saves valuable time. You should not need to look at the reference sheet.
虽然SAT参考框中提供了几何公式，但熟练掌握可以节省宝贵时间。你不应该需要去查看参考表。
| Rectangle: Area = l × w, Perimeter = 2l + 2w | 矩形：面积 = 长 × 宽，周长 = 2(长 + 宽) |
| Triangle: Area = ½ × b × h | 三角形：面积 = ½ × 底 × 高 |
| Circle: Area = πr², Circumference = 2πr | 圆：面积 = πr²，周长 = 2πr |
| Trapezoid: Area = ½(b₁ + b₂) × h | 梯形：面积 = ½(上底 + 下底) × 高 |
The Pythagorean theorem, a² + b² = c², applies to right triangles. Special right triangles — 3-4-5, 5-12-13, and 30-60-90 — appear frequently and can be solved by ratio.
勾股定理 a² + b² = c² 适用于直角三角形。特殊直角三角形——3-4-5、5-12-13 和 30-60-90——频繁出现，可通过比例快速求解。
8. Volume & Surface Area | 体积与表面积
Volume questions often ask you to calculate the capacity of a three-dimensional object or compare volumes after scaling dimensions.
体积题常要求你计算三维物体的容量，或比较缩放尺寸后的体积变化。
| Rectangular prism: V = l × w × h | 长方体：V = 长 × 宽 × 高 |
| Cylinder: V = πr²h | 圆柱：V = πr²h |
| Sphere: V = (4/3)πr³ | 球体：V = (4/3)πr³ |
| Cone: V = (1/3)πr²h | 圆锥：V = (1/3)πr²h |
When all dimensions of a solid are multiplied by a scale factor k, the volume is multiplied by k³. This relationship is a favorite SAT trick for ratio questions.
当一个立体图形的所有维度都乘以缩放因子 k 时，体积乘以 k³。这一关系是SAT比例题中常用的技巧。
9. Coordinate Geometry & Circles in the Plane | 坐标几何与平面圆
Coordinate geometry combines algebra and geometry. Questions may ask about distances, midpoints, and circle equations in the xy-plane.
坐标几何将代数与几何结合。题目可能涉及距离、中点以及xy平面中的圆的方程。
Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)
Circle equation: (x − h)² + (y − k)² = r²
- Distance formula: derived from the Pythagorean theorem.
- 距离公式：由勾股定理推导而来。
- Circle center: (h, k); radius: r.
- 圆心：(h, k)；半径：r。
- is given as x² + y² + Dx + Ey + F = 0, complete the square to find the center and radius.
- 若方程以 x² + y² + Dx + Ey + F = 0 给出，通过配方找到圆心和半径。
Completing the square is one of the most tested algebraic manipulations in SAT coordinate geometry. Practice it until it becomes automatic.
配方是SAT坐标几何中最常考查的代数运算之一。请练习到能够自动完成为止。
10. Trigonometry | 三角函数
SAT trigonometry questions are limited to right-triangle trig and radian measure. The basic ratios are essential.
SAT三角函数题仅限于直角三角形三角比和弧度制。基本比例至关重要。
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.
- 倒数关系：csc θ = 1/sin θ，sec θ = 1/cos θ，cot θ = 1/tan θ。
- Key angle values: sin 30° = ½, cos 60° = ½, tan 45° = 1.
- 常用角度值：sin 30° = ½，cos 60° = ½，tan 45° = 1。
- Radian conversion: π radians = 180°.
- 弧度换算：π 弧度 = 180°。
90° − θ). This complementary-angle relationship appears in several official SAT questions.
对于任意两个和为90°的角度，sin θ = cos(90° − θ)。这一余角关系出现在多道官方SAT题目中。
11. Complex Numbers | 复数
Complex numbers appear in about 2-3 questions per test. The key definition is i² = −1.
复数每次考试约出现2-3题。关键定义是 i² = −1。
- Standard form: a + bi, where a is the real part and b is the imaginary part.
- 标准形式：a + bi，其中 a 为实部，b 为虚部。
- Powers of i: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then the cycle repeats.
- i 的幂：i¹ = i，i² = −1，i³ = −i，i⁴ = 1，然后循环重复。
- Squaring: (a + bi)² = a² + 2abi − b².
- 平方：(a + bi)² = a² + 2abi − b²。
- Conjugate: a − bi. Multiplying by the conjugate eliminates the imaginary part from a denominator.
- 共轭：a − bi。乘以共轭可消除分母中的虚数部分。
When dividing complex numbers, multiply both numerator and denominator by the conjugate of the denominator. This is the most common complex-number operation on the SAT.
复数除法时，分子分母同时乘以分母的共轭复数。这是SAT中最常见的复数运算。
12. Data Analysis & Units | 数据分析与单位换算
Data analysis questions involve scatterplots, line of best fit, and interpreting trends. You may also need to convert between units (e.g., miles to kilometers, gallons to liters).
数据分析题涉及散点图、最佳拟合线和趋势解读。你还可能需要在单位之间进行换算（例如英里与公里、加仑与升）。
- Line of best fit: y = mx + b. Use it to make predictions by substituting x-values.
- 最佳拟合线：y = mx + b。通过代入 x 值进行预测。
- Scatterplot trends: positive correlation (rise together), negative correlation (one rises as the other falls), no correlation (no pattern).
- 散点图趋势：正相关（同升同降）、负相关（一个升一个降）、无相关（无规律）。
- Unit conversion strategy: set up a proportion based on the given equivalence.
- 单位换算策略：根据给定的等价关系列出比例式。
f converting from miles to kilometers, the number should increase since 1 mile ≈ 1.609 km.
单位换算时，检查答案在量纲上是否合理。若从英里换算为公里，数值应变大，因为 1 英里 ≈ 1.609 公里。
Mastering these formulas is the first step to SAT math success. Print this list, review it daily, and test yourself by solving SAT-style problems that require each formula. Repetition builds speed, and speed builds confidence.
掌握这些公式是SAT数学成功的第一步。打印本清单，每日复习，并通过练习需要运用各公式的SAT风格题目来检验自己。重复带来速度，速度带来信心。
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