If you are taking IB Mathematics: Analysis and Approaches (AA), the formula booklet is your best friend during exams. However, walking into a test with 15 pages of formulas can be overwhelming if you don’t know what to look for.
手冊分為五個核心主題：數與代數、函數、幾何與三角學、統計與概率以及微積分。無論你是標準水平（SL）還是高級水平（HL），了解哪些公式是「高產出」的是節省時間和減少壓力的關鍵。
Let’s break down the most critical formulas from each topic.
主題 1：數與代數
等差數列
Formula: or
功能： 求等差數列中任意項或項的和。
何時使用：
- 固定分期的貸款還款
- 每週/每月等額儲蓄
- Any pattern like 5, 9, 13, 17…
如何使用： Identify (first term), (difference), and (term number). Plug into the formula.
等比數列
Formulas:
only if
功能： 模擬重複乘法（增長/衰減）。
何時使用：
- 人口增長
- 放射性衰變
- 汽車價值折舊
- 銀行利息（但複利公式更好）
如何使用： Find and common ratio . For the infinite sum, check.
複利
Formula:
功能： 計算複利投資/貸款的未來價值。
何時使用： 任何按月、按季等複利計算的銀行帳戶、貸款或投資。
如何使用：
- = 現值（初始值）
- = 年利率（%）
- = 每年複利期數
- = 年數
⚠️ Don’t confuse it with a geometric sequence—this one handles intra-year compounding.
對數與指數
Formulas:
功能： 在指數和對數形式之間轉換；將乘法/除法簡化為加法/減法。
何時使用：
- 求解指數 (e.g., )
- 分貝標度（對數）
- pH 標度
- 里氏震級
如何使用： Identify the base. Use the rules to combine or split logs. To solve
rewrite as
.
二項式定理
Formula:
功能： Expands without multiplying manually.
何時使用：
- 求二項式展開中的特定項
- 概率（二項分佈）
- 微積分中的近似
如何使用： For term with , coefficient = . Remember: exponent of is , exponent of is .
主題 2：函數
二次函數對稱軸
Formula:
功能： Finds the vertical line through the vertex of a parabola.
何時使用：
- Finding vertex coordinates
- Maximizing/minimizing quadratic functions (e.g., profit, projectile height)
如何使用： Given , plug and into the formula.
Discriminant
Formula:
功能： Tells you the nature of roots.
何時使用：
- Before solving a quadratic
- Determining if a quadratic intersects the x-axis
How to interpret:
- : two real distinct roots
- : one repeated root (tangent)
- : no real roots (complex)
Sum & Product of Polynomial Roots
Formulas: Sum of roots = Product of roots =
功能： Relates roots to coefficients without solving.
何時使用：
- Forming a quadratic from its roots
- Checking roots quickly
- Symmetric root problems
如何使用： For : sum = , product = . For higher degree, match terms.
Topic 3: Geometry & Trigonometry
Sine Rule
Formula:
何時使用：
- AAS (two angles, one side)
- ASA (two angles, included side)
- SSA (ambiguous case — check)
如何使用： Set up proportion with the side-angle pair you know. Solve for unknown.
Cosine Rule
Formulas:
何時使用：
- SSS (all three sides)
- SAS (two sides, included angle)
如何使用： First version: find side. Second version: find angle.
Area of a Triangle (Trig)
Formula:
何時使用： You know two sides and the included angle (SAS).
如何使用： Multiply: 0.5 × side a × side b × sine of the angle between them.
Arc Length & Sector Area
Formulas:
何時使用： Any circle problem where the angle is in radians (convert if needed).
如何使用： Ensure is in radians. Plug and solve.
Pythagorean & Double Angle Identities
Formulas:
何時使用：
- Simplifying trig expressions
- Solving trig equations
- Integrating trig functions
如何使用： Replace with etc. Use double angles to reduce powers.
Topic 4: Statistics & Probability
Mean from Grouped Data
Formula:
何時使用： Data is grouped into intervals (e.g., 0–10, 10–20).
如何使用： = midpoint of interval. Multiply by frequency, sum, divide by total frequency.
Conditional Probability
Formula:
何時使用： Probability of A given that B has occurred.
如何使用： Identify the reduced sample space (B). Find intersection over B’s probability.
Binomial Distribution
Formulas: Mean Variance
何時使用： Fixed number of trials (), two outcomes (success/fail), constant probability (), independent trials.
如何使用： Check BINS: Binary, Independent, Number fixed, Same probability.
Normal Distribution (Standardization)
Formula:
何時使用： Finding probabilities for normal distributions using z-tables.
如何使用： Convert raw score to z-score, then look up probability.
Bayes’ Theorem (HL only)
Formula:
何時使用： Updating probability based on new evidence (medical tests, spam filters).
如何使用： Draw a tree diagram first. Then plug into formula.
Topic 5: Calculus
Basic Derivatives (SL)
| Function | Derivative |
|---|
何時使用： Finding slope of tangent, rates of change, optimization.
如何使用： Apply power rule, trig rules, or exponential rules directly.
Chain, Product, Quotient Rules
- Chain:
*Use for composite functions like* *.*
- Product:
*Use for .*
- Quotient:
*Use for .*
Basic Integrals (SL)
何時使用： Finding area under curve, reversing differentiation, solving differential equations.
如何使用： Add 1 to power, divide by new power. Don’t forget for indefinite integrals.
Kinematics (SL)
- Acceleration:
- Displacement:
- Distance:
何時使用： Motion problems (particle moving along a line).
如何使用：
- 位移 = 最終位置 − 初始位置（可以為負）
- 距離 = 總路徑長度（始終為正 — 對速度的絕對值積分）
分部積分（僅 HL）
Formula:
何時使用： Integrating products of functions, e.g., , . ,
How to choose and Use LIATE rule: Log → Inverse trig → Algebraic → Trig → Exponential (u = first in LIATE, dv = rest)
最終建議：如何掌握公式手冊
- Don’t memorize — practice locating. Under exam pressure, you need to flip to the right page in seconds.
- Annotate your booklet. Write small reminders, such as “use for compound interest,” next to the formula.
- Practice without the booklet first, then with it.This builds both memory and lookup speed.
- Know what’s NOT in the booklet. E.g., unit circle, calculator commands, common derivatives/integrals — memorize those.
- For HL: Focus on the “HL only” sections (complex numbers, vectors, Bayes, Maclaurin, parts). They appear in Papers 2 and 3.
公式手冊不是教科書 — it’s a 工具. 你越了解*為什麼*和*何時*使用每個公式，考試就越不令人畏懼。
以 All Round 的方式學習
掌握 IB Math AA SL 和 HL 的基本公式，清晰解讀官方公式手冊，自信地應用關鍵方程式，更快地解決複雜問題，並通過詳細的解釋和範例提升你的考試表現。如果你需要更多指導，我們邀請你與 All Round Education Academy 聯繫。我們的專業團隊隨時為你的學業目標提供支持。如需更多資訊，請聯繫我們：[email protected] 或 +852 6348 8744。