📚 IB Math: Casio fx-CG20 Key Functions Explained | IB 数学：casio fx-cg20 知识点精讲
The Casio fx-CG20 is a powerful graphing calculator widely approved for IB Mathematics examinations. Mastering its functions can save time, reduce errors, and enhance problem-solving efficiency across all topics from algebra to calculus. This guide walks you through the essential features every IB student should know, with step-by-step explanations and practical examples tailored to the IB syllabus.
Casio fx-CG20 是一款功能强大的图形计算器，被广泛允许在 IB 数学考试中使用。熟练掌握它的各项功能可以节省时间、减少错误，并能提升从代数到微积分各知识点的解题效率。本篇指南将带你逐一了解每位 IB 学生都应掌握的核心功能，每一步都配有详细说明和紧扣 IB 考纲的实际示例。
1. Getting Started with the fx-CG20 | fx-CG20 入门与基本设置
Before diving into advanced features, it is essential to familiarise yourself with the calculator’s layout and basic settings. The fx-CG20 runs on a menu-driven interface: press the MENU key to access icons for Run-Matrix, Statistics, Graph, Equation, and many other modes. Use the arrow keys to navigate and EXE to select.
在深入高级功能之前，熟悉计算器的布局和基本设置至关重要。fx-CG20 采用菜单驱动界面：按下 MENU 键即可进入包含 Run-Matrix、Statistics、Graph、Equation 等图标的模式选择。使用方向键导航，按 EXE 执行选择。
A crucial initial step is to set the calculator’s angle unit (degrees or radians) and display format. Go to SHIFT → SET UP, then choose Angle: Deg for degrees or Rad for radians. For IB exams, you will frequently switch between these, especially in trigonometric and calculus questions. Also ensure the Input/Output mode is set to Math for natural textbook display of fractions and radicals.
关键的第一步是设定计算器的角度单位（角度制或弧度制）和显示格式。进入 SHIFT → SET UP，然后在 Angle 选项中选择 Deg 表示角度，或 Rad 表示弧度。在 IB 考试中，三角函数和微积分题目会频繁要求你在两者之间切换。同时还要确保输入/输出模式设为 Math，以便以自然书写方式显示分数和根式。
2. Run-Matrix Mode: The Calculation Engine | Run-Matrix 模式：计算引擎核心
Run-Matrix mode is the fx-CG20’s main screen for performing quick calculations, evaluating expressions, and solving simple equations numerically. It functions like a scientific calculator but retains editable command lines so you can correct mistakes or reuse expressions. Access it via the icon with the ‘4-lines’ symbol in the Main Menu.
Run-Matrix 模式是 fx-CG20 的主界面，用于快速计算、求表达式的值以及数值求解简单方程。它像科学计算器一样工作，但保留可编辑的命令行，方便修改错误或重复使用表达式。通过主菜单中“四行”符号的图标即可进入。
To evaluate a function like f(x) = 2x² − 5x at x = 3, type the expression directly and press EXE. You can also store values in variables using the STO (→) button: assign 3 → A, then type 2A² − 5A and EXE. The clipboard function (OPTN → CLIP) helps you copy results. For quick numerical solutions, use the Solve feature found under OPTN → CALC → Solve, which applies Newton’s method to find a root near your initial guess.
若要计算函数 f(x) = 2x² − 5x 在 x = 3 时的值，直接输入表达式并按 EXE 即可。你也可以使用 STO (→) 按钮将值存入变量：输入 3 → A，再输入 2A² − 5A 并按 EXE。剪贴板功能（OPTN → CLIP）可帮助复制结果。对于快速数值求解，可使用 OPTN → CALC → Solve 中的求解功能，它应用牛顿法在你给出的初始猜测值附近寻找根。
3. Graph Mode: Visualising Functions | Graph 模式：函数图像可视化
Graph mode is perhaps the most used feature for IB Math Analysis & Approaches and Applications & Interpretation. Enter Graph mode from the Main Menu, then input up to 20 functions in the form Y1=, Y2=, etc. Press DRAW to see the graphs. The V-Window (view window) settings under SHIFT → F3 determine the axis ranges; Auto-scale (F3 → Auto) can quickly frame key features.
Graph 模式或许是 IB 数学分析与方法和应用与解释中最常用的功能。从主菜单进入 Graph 模式，然后以 Y1=、Y2= 等形式输入多达 20 个函数。按 DRAW 查看图像。SHIFT → F3 下的 V-Window（视窗）设置决定坐标轴范围；自动缩放（F3 → Auto）可快速框定关键特征。
This mode allows you to find intersections, roots, maxima, minima, and y-intercepts with precision. After drawing the graph, press G-Solve (F5) and select the desired operation: ROOT for zeros, MAX/MIN for extrema, Y-ICEPT for the y-intercept, and ISCT for intersection of two graphs. For intersection, you need at least two function graphs drawn; the cursor will jump to the solution, and the coordinates are displayed.
该模式能够精确求取交点、零点、最大值、最小值和 y 轴截距。绘制图像后，按 G-Solve (F5) 并选择所需操作：ROOT 求零点，MAX/MIN 求极值，Y-ICEPT 求 y 轴截距，ISCT 求两条图像的交点。对于交点，至少需要绘制出两个函数图像；光标会跳转到解的位置并显示坐标。
4. Equation Mode: Solving Polynomials and Systems | Equation 模式：求解多项式与方程组
Equation mode handles linear equations, quadratic/cubic polynomials, and systems of linear equations directly. From the Main Menu, choose the Equation icon. You will see options: Simultaneous Linear, Polynomial (degree 2 to 6), and Solver. For a quadratic, select Polynomial, then degree 2, and input coefficients a, b, c; press SOLVE to get both real and complex roots automatically.
Equation 模式可直接处理一次方程、二次/三次多项式以及线性方程组。从主菜单选择 Equation 图标。你将看到以下选项：Simultaneous Linear（联立方程）、Polynomial（2 至 6 次多项式）和 Solver（求解器）。对于二次方程，选择 Polynomial，再选 degree 2，输入系数 a、b、c；按 SOLVE 即可自动得到所有实根与复根。
For a system like 2x + 3y = 7 and 5x − y = 8, select Simultaneous Linear, enter the number of unknowns (2), then input the coefficients in matrix form. The calculator returns the unique solution, including cases with infinite or no solutions, displaying a message when the determinant is zero. This is invaluable for solving geometric intersection problems or balanced chemical equations (as in IB Chemistry).
对于如 2x + 3y = 7 和 5x − y = 8 的方程组，选择 Simultaneous Linear，输入未知数个数 (2)，然后以矩阵形式输入系数。计算器会返回唯一解，即便有无穷多解或无解的情况也能给出相应提示（当行列式为零时）。这对解决几何交点问题或配平化学方程式（IB 化学中）非常有价值。
5. Table Mode: Generating Function Values | Table 模式：生成函数值表
Table mode is perfect for exploring function behaviour, spotting patterns, and checking limits or asymptotes. Enter Table mode, input a function in Y1=, then define the table range with Start, End, and Step values. The calculator generates a two-column table of x and f(x). You can scroll through the values and even trace discontinuities.
Table 模式非常适合探索函数行为、发现规律以及检查极限或渐近线。进入 Table 模式，在 Y1= 中输入函数，然后用 START、END 和 STEP 值定义表格范围。计算器会生成 x 与 f(x) 两列表格。你可以滚动浏览数值并追踪间断点。
This is especially useful for verifying solutions to equations or inequalities. For instance, when solving a trigonometric equation, you can generate a table between 0 and 2π with a step of π/6 to see where sin(x) takes certain values. The table also supports displaying derivative values if you set the Dual Screen mode to Graph & Table, enabling deeper analysis of rates of change alongside function values.
这在验证方程或不等式的解时尤其有用。例如，在解三角方程时，你可以生成 0 到 2π 之间步长为 π/6 的表格，查看 sin(x) 在何处取某一特定值。如果将双屏模式设为 Graph & Table，该表格还支持显示导数值，从而将变化率与函数值一起进行更深入的分析。
6. Statistics Mode: Data Handling and Regression | Statistics 模式：数据处理与回归分析
Statistics mode is a critical tool for the Applications & Interpretation course and the Internal Assessment. Enter Stat mode, then input data into List 1 and List 2 (or more lists). The calculator can compute 1-variable statistics (mean, median, standard deviation, quartiles) and 2-variable statistics (correlation, regression). Press CALC (F2) then 1-VAR or 2-VAR to see results.
Statistics 模式是应用与解释课程以及内部评估（IA）的关键工具。进入 Stat 模式，将数据输入 List 1 和 List 2（或更多列表）。计算器可计算单变量统计量（均值、中位数、标准差、四分位数）和双变量统计量（相关性、回归）。按 CALC (F2)，再选择 1-VAR 或 2-VAR 查看结果。
For regression, after entering paired data, press CALC, then REG (F3). Choose from linear, quadratic, cubic, exponential, logarithmic, sinusoidal, and logistic models. The fx-CG20 displays the regression equation, correlation coefficient r, and coefficient of determination r². It can also overlay the regression curve on a scatter plot (GRAPH → SET → Graph Type: Scatter, then DRAW) to visually assess fit. This function is heavily tested in exam Paper 2 and essential for IA data analysis.
对于回归分析，输入成对数据后，按 CALC，然后 REG (F3)。可选模型包括线性、二次、三次、指数、对数、正弦和逻辑回归。fx-CG20 会显示回归方程、相关系数 r 和决定系数 r²。它还可以将回归曲线叠加在散点图上（GRAPH → SET → Graph Type: Scatter，然后 DRAW），以直观评估拟合优度。这一功能在考试卷二和 IA 数据分析中频繁考查。
7. Probability Distributions: Normal, Binomial, Poisson | 概率分布：正态、二项、泊松
Probability calculations are effortless with the built-in distribution wizard. From the main menu, select the Stat icon, then choose DIST (F5) to access the distribution menu. The available distributions include Normal (NORM), Binomial (BINM), Poisson (POISN), Geometric (GEO), Hypergeometric (H.GEO), and more. Each offers Probability Density (P.D), Cumulative Distribution (C.D), and Inverse (Inv) options.
借助内置分布向导，概率计算变得毫不费力。从主菜单选择 Stat 图标，然后按 DIST (F5) 进入分布菜单。可用分布包括正态 (NORM)、二项 (BINM)、泊松 (POISN)、几何 (GEO)、超几何 (H.GEO) 等。每种分布都提供概率密度 (P.D)、累积分布 (C.D) 和逆累积 (Inv) 选项。
To find P(X ≤ 7) for X~Bin(12, 0.4), go to DIST → BINM → Bcd, then input Data: Variable, x: 7, Numtrial: 12, p: 0.4, and execute. The result appears instantly. For inverse normal, which finds the z-value for a given tail probability, use NORM → InvN. This is indispensable for confidence interval work and hypothesis testing in the statistics component of the IB course.
计算 X~Bin(12, 0.4) 时 P(X ≤ 7)，进入 DIST → BINM → Bcd，然后输入 Data: Variable, x: 7, Numtrial: 12, p: 0.4，执行即可。结果立即显示。对于逆正态分布，即求给定尾部概率对应的 z 值，使用 NORM → InvN。这在 IB 统计部分的置信区间和假设检验中不可或缺。
8. Calculus Operations: Numerical Integration and Derivatives | 微积分运算：数值积分与求导
The fx-CG20 performs numerical differentiation and definite integration directly from the Run-Matrix and Graph modes. In Run-Matrix mode, press OPTN → CALC, then select d/dx for derivative at a point or ∫dx for definite integral. For d/dx, you specify the function, the variable, and the x-value where the derivative is evaluated.
fx-CG20 可直接在 Run-Matrix 和 Graph 模式下进行数值微分和定积分计算。在 Run-Matrix 模式，按 OPTN → CALC，然后选择 d/dx 求某一点的导数，或选择 ∫dx 求定积分。对于 d/dx，你需要指定函数、变量以及求导点的 x 值。
To compute the definite integral of f(x) = x² sin(x) from 0 to π, type ∫(X² sin X, 0, π) using the ∫ template. Ensure the calculator is in radian mode for trigonometric integrands. In Graph mode, you can use G-Solve → ∫dx to shade the area under a curve between two x-values and display the signed area. For the area between two curves, graph both, then use G-Solve → Intersect to find bounds, and then set upper and lower functions in the integration setup. This visual approach aids understanding of the Fundamental Theorem of Calculus.
要计算 f(x) = x² sin(x) 从 0 到 π 的定积分，使用 ∫ 模板输入 ∫(X² sin X, 0, π)。对于含有三角函数的被积函数，务必使计算器处于弧度模式。在 Graph 模式下，可使用 G-Solve → ∫dx 对两个 x 值之间的曲线下方区域进行着色并显示有向面积。要求两条曲线之间的面积，先绘制两者，再使用 G-Solve → Intersect 求出界限，然后在积分设置中指定上下函数。这种可视化方法有助于理解微积分基本定理。
9. Matrices and Vectors: Linear Transformations | 矩阵与向量：线性变换运算
Matrix operations are accessible through the Run-Matrix mode. Press OPTN → MAT to bring up the matrix menu, where you can define matrices (Mat A, Mat B, …) with dimensions up to 999 × 999. After storing matrices, you can perform addition, subtraction, multiplication, determinant (det), inverse (⁻¹), transpose (Trn), and row echelon form (Ref/Rref).
矩阵运算可通过 Run-Matrix 模式进行。按 OPTN → MAT 调出矩阵菜单，在此定义维度高达 999 × 999 的矩阵（Mat A、Mat B……）。存储矩阵后，即可进行加、减、乘、行列式 (det)、逆矩阵 (⁻¹)、转置 (Trn) 和行阶梯形 (Ref/Rref) 等运算。
For IB, the rref command is particularly useful for solving systems of linear equations and determining linear independence. Enter the augmented matrix, then run Rref Mat A to obtain the reduced row echelon form. The last column gives the solution. Vector dot products, cross products, and angles can be computed by storing vectors as 3 × 1 matrices and using the dotP and crossP commands found under OPTN → MAT → VCT. This streamlines questions on planes, distances, and intersections in the Geometry and Trigonometry topic.
在 IB 中，rref 命令对解线性方程组和判断线性相关/无关尤为有用。输入增广矩阵，然后运行 Rref Mat A 即可得到简化行阶梯形矩阵，最后一列即为解。向量的点积、叉积和夹角可以通过将向量存储为 3 × 1 矩阵，并使用 OPTN → MAT → VCT 下的 dotP 和 crossP 命令来计算。这大大简化了几何与三角学专题中关于平面、距离和交点的问题。
10. Sequences and Series: Recursion and Summation | 序列与级数：递推与求和
The Recursion mode (RECUR) is designed for exploring sequences defined explicitly or recursively. Enter the mode from the Main Menu, then define a sequence: an+1 = Aan + B (linear recursion) or an = expression in n (explicit). You can set two sequences and display them in a table or graph. The Table view shows term numbers and values, while the Graph view plots points (n, an).
Recursion 模式（RECUR）专为探索显式或递推定义的序列而设计。从主菜单进入该模式，然后定义序列：an+1 = Aan + B（线性递推）或 an = 含 n 的表达式（显式）。你可以设置两个序列并以表格或图形方式显示。表格视图显示项数和项值，图形视图则绘制点 (n, an)。
For IB exam-type questions on compound interest, loan repayments, or population growth, use the recursive definition with the initial term and rule. The calculator can also compute the sum of terms using the Σ command in Run-Matrix mode: press OPTN → CALC → Σ, then input the expression, variable, start, and end values. This saves time on arithmetic series and confirms closed-form sum formulas.
对于 IB 考试中关于复利、贷款还款或人口增长的典型问题，可使用带初始项和规则的递推定义。计算器还可使用 Run-Matrix 模式中的 Σ 命令求和：按 OPTN → CALC → Σ，然后输入表达式、变量、起始值和终止值。这能节省计算算术级数的时间，并验证求和公式的封闭形式。
11. Programming: Custom Scripts for Repetitive Tasks | 编程：定制脚本处理重复任务
The fx-CG20 supports a BASIC-like programming language that allows you to automate formulas or create reusable tools. From the Main Menu, choose the Program icon (PRGM). Create a new program, give it a name, and write a sequence of commands using the calculator’s intuitive command menus. For example, a program can prompt for inputs, perform quadratic discriminant analysis, and display the nature of roots.
fx-CG20 支持一种类 BASIC 的编程语言，可用于自动计算公式或创建可重复使用的工具。从主菜单选择 Program 图标 (PRGM)。创建新程序，为其命名，并使用计算器直观的命令菜单编写一系列命令。例如，一个程序可以提示输入、进行二次判别式分析并显示根的性质。
Programming is not required in the IB syllabus, but it can be a powerful aid for your IA, particularly when running simulations or iterating numerical methods like Euler’s method for differential equations. The language includes loops (For, While), conditional statements (If-Then-Else), and graphical output commands. Even simple programs can speed up checks and reduce repetitive button pressing during revision.
编程并非 IB 大纲的硬性要求，但对 IA 特别有帮助，尤其是在运行模拟或迭代数值方法（如解微分方程的欧拉法）时。该语言包含循环 (For, While)、条件语句 (If-Then-Else) 和图形输出命令。即使简单的程序也能加快检查速度，减少复习时重复按键的次数。
12. Exam Tips and Efficient Workflow | 考试技巧与高效工作流程
To maximise your performance with the fx-CG20 in the IB exam, develop a consistent routine. Always start by checking angle units. Use the clipboard to copy long decimals for further calculations rather than retyping. Store intermediate results as variables (A, B, C, …) to retain precision. Familiarise yourself with the RESET procedure (System Manager), as you may be asked to clear memory before the exam.
要在 IB 考试中充分运用 fx-CG20 的性能，需形成一套稳定的操作流程。每次开始前检查角度单位。使用剪贴板复制长小数以便进一步计算，而非重新键入。将中间结果存入变量（A、B、C……）以保持精度。熟悉重置步骤（System Manager），因为考前可能会要求清除内存。
Use the calculator’s multiple modes cleverly: sketch graphs to verify algebraic solutions, use tables to check factorisation, and employ the equation solver for transcendental equations that cannot be solved analytically. Remember that while the calculator shows many digits, final answers should be given to three significant figures unless otherwise stated. Practise these techniques regularly so that button presses become instinctive, letting you focus on mathematical reasoning rather than device operation.
巧妙运用计算器的多种模式：绘制图像验证代数解，使用表格检查因式分解，用方程求解器处理无法解析求解的超越方程。记住，尽管计算器显示多达许多位数字，最终答案除特别说明外应保留三位有效数字。定期练习这些技巧，让按键操作成为本能，从而将注意力集中于数学推理而非设备操作。
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