📚 Mastering the TI-84 for IB Applications and Interpretation HL | IB AI HL 中 TI-84 计算器高效使用指南
The TI-84 family of graphing calculators is an essential tool for IB Mathematics: Applications and Interpretation HL. Because AI HL allows a graphic display calculator in every examination paper, calculator fluency is not an optional extra - it is part of the assessed skill set. This guide covers the most common tasks, settings, and exam techniques that will help you use your calculator accurately and efficiently, from basic MODE configuration to numerical calculus, statistical regression, probability distributions, finance and matrices.
TI-84 系列图形计算器是 IB 数学：应用与解释 HL 的重要工具。由于 AI HL 的每一份试卷都允许使用图形计算器，熟练操作计算器并非"额外技能"，而是被考核能力的一部分。本指南涵盖最常见的任务、设置和考试技巧，帮助你准确高效地使用计算器——从基础的 MODE 设置，到数值微积分、统计回归、概率分布、金融计算与矩阵运算。
1. Calculator Basics and Essential Settings | 计算器基础与必要设置
Before any exam question, set the MODE menu correctly. Use Float for full precision in working, but switch to 3 decimal places only if the question asks for rounding. Ensure angle measure is in radians for calculus and trigonometry, and choose FUNCTION graphing mode. In AI HL the overwhelming majority of calculus and trigonometric modelling questions are set in radians, so a calculator left in Degree mode will silently produce wrong answers on the Paper 3 technology questions.
在任何考试题之前，先正确设置 MODE 菜单。计算过程中使用 Float 保持完整精度，仅当题目要求四舍五入时才切换到 3 位小数。确保角度单位为弧度用于微积分和三角学，并选择函数绘图模式。在 AI HL 中，绝大多数微积分与三角函数建模题都使用弧度，若计算器仍停留在 Degree 模式，Paper 3 的技术类题目会悄悄出错。
The key MODE settings to check, and the reason each one matters:
需要核查的关键 MODE 设置，以及每一项为何重要：
- Float / 3 / 4 … 9: Float keeps all internal digits; the fixed decimal settings round the display only. For IB, answer to 3 significant figures unless the question specifies otherwise - but do the working in Float. | Float 保留全部内部位数；固定小数位只改变显示。IB 一般要求 3 位有效数字作答，但运算是用 Float 完成。
- Radian / Degree: Set Radian for calculus, trigonometric modelling, and anything involving arc length or sector area with real-number angles. Degree is only for explicitly stated degree problems. | 涉及微积分、三角建模、弧长与扇形面积时用弧度。只有题目明确给度数时才用 Degree。
- FUNCTION / PARAMETRIC / POLAR / SEQ: Keep FUNCTION for almost all AI HL work; switch to SEQ only when graphing sequences. | 绝大多数 AI HL 题目用 FUNCTION；只有画数列图时才切到 SEQ。
- Real / a+bi / re^θi: Stay in Real unless the question involves complex numbers, which appear in AI HL. | 除非题目涉及复数（AI HL 有），否则保持 Real。
- Full / Horiz / G-T: Full screen for graphing; G-T is occasionally handy when you want the table beside the graph. | 绘图用 Full；需要边看图边看表时可用 G-T。
A five-second MODE check at the start of every practice paper builds the habit that saves whole marks in the real exam.
每次做练习卷前花五秒检查 MODE，可以养成在正式考试中直接救回整道题分数的习惯。
2. Memory, Reset and Pre-Exam Housekeeping | 内存清理与考前准备
ce values from interfering with new work. Nothing is more frustrating than an old Y= entry being graphed on top of your new model, or a stale FV value in the TVM Solver flipping the sign of your finance answer.
在模拟考试或正式考试前，按 2nd + MEM，然后 7: Reset，选择 RAM 或 Defaults 清除内存。这可以防止旧的函数、列表或金融数值干扰新计算。没有什么比旧的 Y= 表达式叠在你的新模型上绘图，或 TVM Solver 里残留的 FV 数值把金融答案的符号翻转更令人沮丧了。
Check the diagnostic setting so that regression output includes r and r squared. Go to 2nd + CATALOG, scroll or jump to DiagnosticOn, and press ENTER twice. Without diagnostics switched on, LinReg, ExpReg and other regression commands will not display the correlation coefficient, and AI HL questions frequently ask you to interpret r or r squared.
检查诊断设置，让回归输出包含 r 与 r²。进入 2nd + CATALOG，滚动或跳转到 DiagnosticOn，按两次 ENTER。若未开启诊断，LinReg、ExpReg 等回归命令不会显示相关系数，而 AI HL 常要求你解释 r 或 r² 的意义。
Housekeeping checklist before the exam:
考前整理清单：
- Reset RAM, then re-enable DiagnosticOn and re-set Radian mode. | 先清 RAM，再重新开启 DiagnosticOn 并设回弧度模式。
- Delete unused lists in STAT EDIT to avoid confusion with data entry. | 在 STAT EDIT 中删除无用列表，避免数据输入混淆。
- Insert four fresh AAA batteries and know where the backup battery is. | 换上四节新 AAA 电池，并知道备用电池位置。
- Confirm your model is an allowed graphic display calculator and not a CAS model. | 确认机型是允许的图形计算器，而不是 CAS 机型。
3. What the AI HL Papers Actually Allow | AI HL 试卷实际允许什么
In IB Mathematics: Applications and Interpretation HL, a graphic display calculator is permitted in Paper 1, Paper 2 and Paper 3. Paper 3 is the extended "technology" paper and carries the largest per-question weight, so the calculator is effectively mandatory there. Crucially, when a question states "using technology" or asks for a value found graphically or numerically, you are expected to show the setup, not just the number.
在 IB 数学：应用与解释 HL 中，Paper 1、Paper 2 和 Paper 3 均允许使用图形计算器。Paper 3 是扩展型"技术"卷，单题分值最高，因此计算器在那里几乎是必需工具。关键在于，当题目写明"using technology"，或要求用图像法/数值法求值时，你应该展示设定过程，而不仅是写出数字。
Command terms and their calculator implications:
指令词及其计算器含义：
| Command term | 指令词 | What to write | 应写内容 |
| Write down | 写出 | No working required; a calculator result alone is accepted. | 无需过程，计算器结果可直接接受。 |
| Find / Calculate | 求 / 计算 | Show the equation, function or command used, then the answer. | 写出所用方程、函数或命令，再给答案。 |
| Solve | 解 | State the method (graphical intersection, Solver, PlySmlt2) and the roots. | 说明方法（图像交点、Solver、PlySmlt2）并给出根。 |
| Sketch | 作图 | Axes, intercepts and shape matter; a calculator screenshot is not a sketch. | 关键在坐标轴、截距与形状；计算器截图不等于作图。 |
Always state the window or the command you used when a result is obtained numerically - that is what earns the method marks.
当结果是数值求得时，务必写明所用窗口或命令——这才是拿到方法分的关键。
4. Entering Functions, Tables and the Y= Menu | 输入函数、表格与 Y= 菜单
he fraction template (ALPHA + Y=) rather than a slash when a numerator or denominator is a sum, otherwise you will introduce bracketing errors.
几乎每一道图像或数值题都从 Y= 开始。按 Y=，用 CLEAR 清除遗留项，再用 X,T,θ,n 键输入函数。当分子或分母是多项式时，用分式模板（ALPHA + Y=）而不是斜杠，否则容易产生括号错误。
For a table of values, press 2nd + GRAPH (TABLE). Use 2nd + WINDOW (TBLSET) to control TblStart and ΔTbl, or set Indpnt: Ask to enter your own x-values. This is the fastest way to build a table for a modelling question, estimate a limit, or generate data for a difference table.
要看数值表，按 2nd + GRAPH（TABLE）。用 2nd + WINDOW（TBLSET）控制 TblStart 与 ΔTbl，或设 Indpnt: Ask 以便自己输入 x 值。这是为建模题列表、估算极限或构造差分表的最快方法。
Practical tips for entering expressions correctly:
正确输入表达式的实用技巧：
- ( ) brackets every time a negative sign appears inside a power or a function. | 每当负号出现在幂或函数内部时，都要加括号。
- Use ^ for powers and the ˣ√ template for roots; ³√x is safer than x1/3 for negative x. | 幂用 ^，根用 ˣ√ 模板；对负数 x 而言 ³√x 比 x1/3 更安全。
- The ex key is 2nd + LN; log base 10 is LOG, natural log is LN. | e 的指数键是 2nd + LN；常用对数用 LOG，自然对数用 LN。
- Store long constants with STO→ to reuse them exactly and avoid rounding. | 用 STO→ 存储常数，精确复用，避免舍入误差。
5. Graphing, Windows and Reading a Model | 绘图、窗口与读图
The default window is rarely the right window. Press WINDOW and set Xmin, Xmax, Ymin, Ymax based on the context - a population model may need Ymax in the thousands, while a trigonometric model needs Xmax around 4π or more. ZOOM 0: ZDecimal is useful for trigonometric graphs, ZOOM 6: ZStandard for quick checks, and ZOOM 9: ZOOMStat resets the window to fit your statistical data.
默认窗口几乎从来不是合适的窗口。按 WINDOW，根据情境设置 Xmin、Xmax、Ymin、Ymax——人口模型可能需要 Ymax 达数千，而三角模型需要 Xmax 约 4π 或更大。ZOOM 0: ZDecimal 适合三角图像，ZOOM 6: ZStandard 用于快速检查，ZOOM 9: ZOOMStat 可把窗口设为契合统计数据。
mum and 4: maximum for turning points, 5: intersect for solving f(x) = g(x), and 7: ∫f(x)dx for a graphical definite integral. Each command asks you for a Left Bound and Right Bound, so move the cursor so the feature lies between them.
2nd + CALC 菜单是拿到大部分图像分的地方。用 1: value 求某点的函数值，2: zero 求根与 x 轴截距，3: minimum 与 4: maximum 求极值点，5: intersect 解 f(x) = g(x)，7: ∫f(x)dx 做图像法定积分。每个命令都会问 Left Bound 与 Right Bound，因此要把游标移动使目标特征落在二者之间。
A reliable graph-reading routine:
可靠的读图流程：
- Identify the mathematical question behind the picture: intercept, intersection, extremum, area or asymptote. | 先判断图像背后的数学问题：截距、交点、极值、面积或渐近线。
- Set a window that shows the feature with room on both sides. | 设定窗口，使目标特征两侧都留有余量。
- Use the matching CALC command rather than eyeballing coordinates. | 使用对应的 CALC 命令，而不是目测坐标。
- Round only at the end, to 3 significant figures unless told otherwise. | 只在最后按 3 位有效数字（除非另有说明）四舍五入。
6. Solving Equations Numerically | 数值求解方程
LC 5: intersect or 2: zero. Algebraically-numerically, use MATH 0: Solver, which finds a root near a starting guess you supply. For polynomials, use APPS 9: PlySmlt2 (on the TI-84 Plus CE and recent OS versions) to get all real and complex roots at once.
TI-84 上有三条可靠路径。图像法：把左边输入 Y₁、右边输入 Y₂（或整理为 Y₁ = 0），再用 2nd + CALC 5: intersect 或 2: zero。代数-数值法：用 MATH 0: Solver，它会在你给定的初值附近找根。多项式：用 APPS 9: PlySmlt2（TI-84 Plus CE 及较新系统）一次性得到全部实根与复根。
ning Solver, and set bound to {-1E99, 1E99}. A poor guess can return a different root, or the message "no sign change", which usually means the equation has no solution in the bracketed interval.
使用 Solver 时，务必先存一个合理初值。若预计根在 x = 12 附近，就先令 X = 12，再打开 Solver，并把 bound 设为 {-1E99, 1E99}。糟糕的初值可能返回另一个根，或提示 "no sign change"，这通常意味着方程在该区间无解。
Typical AI HL equations and the best route:
典型 AI HL 方程与最佳路径：
- Exponential growth equal to a constant: intersect, because it also lets you read the graph. | 指数增长等于常数：用 intersect，因为还能读图。
- Trigonometric equation in a given interval: graph with Xmax set to the interval end, then find all zeros. | 给定区间上的三角方程：把 Xmax 设为区间端点作图，再找全部零点。
- Cubic or quartic: PlySmlt2 for speed, zero for method marks. | 三次或四次：PlySmlt2 求快，zero 便于拿方法分。
- Equation with no closed form, such as x = cos x: Solver. | 无解析解的方程，如 x = cos x：用 Solver。
7. Calculus: Numerical Derivatives and Definite Integrals | 微积分：数值导数与定积分
ence quotient with step 0.001. Option 9: fnInt(f(x), X, a, b) gives a definite integral, computed by Gauss-Kronrod quadrature. Graphically, 2nd + CALC 6: dy/dx finds the gradient at a point on the curve, and 7: ∫f(x)dx shades and evaluates an area.
TI-84 上 AI HL 的微积分工具在 MATH 菜单。选项 8: nDeriv(f(x), X, a) 给出 x = a 处的数值导数，使用步长为 0.001 的对称差商。选项 9: fnInt(f(x), X, a, b) 给出定积分，采用 Gauss-Kronrod 求积法。图像法下，2nd + CALC 6: dy/dx 求曲线上某点的斜率，7: ∫f(x)dx 会阴影并计算面积。
Key relationships you should be able to move between instantly:
你应当能瞬间切换的关键关系：
f′(a) ≈ [f(a + h) − f(a − h)] ÷ (2h), h = 0.001
∫ₐᵇ f(x) dx = F(b) − F(a) = 曲线下带符号面积
Remember that fnInt returns a signed area: regions below the x-axis contribute negatively. When a question asks for total area enclosed, split the interval at each zero and integrate each piece with the absolute value, or use fnInt of |f(x)| if your OS supports it through the abs( command in the MATH NUM menu.
记住 fnInt 给出的是带符号面积：x 轴下方的区域贡献负值。当题目要求"总围成面积"时，须在每个零点处拆分区间并分别取绝对值，或用 MATH NUM 菜单中的 abs( 命令对 |f(x)| 积分。
Accuracy notes for exam work:
考试作答的准确性提示：
- Write the integral you evaluated, not just the number: "fnInt(f, X, 0, 5) = 12.7". | 写出你计算的积分式，而不仅是数字："fnInt(f, X, 0, 5) = 12.7"。
- Never round intermediate integrals used later in a chain calculation. | 后续链式计算中用到的中间积分值绝不能先舍入。
- If nDeriv gives an odd value, suspect a discontinuity or a cusp at that point. | 若 nDeriv 给出异常值，怀疑该点存在间断或尖点。
8. Statistics: Lists, Summary Statistics and Regression | 统计：列表、描述统计与回归
Enter data through STAT 1: Edit. Put the explanatory variable in L₁ and the response in L₂, then leave the cursor on the list header and press CLEAR ENTER to empty a list cleanly. STAT 1: 1-Var Stats gives mean, standard deviation (use Sx for a sample, σx for a population), median, quartiles and the five-number summary in one screen.
通过 STAT 1: Edit 输入数据。自变量放 L₁，因变量放 L₂；把游标停在列表标题上按 CLEAR ENTER 可干净清空列表。STAT 1: 1-Var Stats 一屏给出均值、标准差（样本用 Sx，总体用 σx）、中位数、四分位数与五数概括。
For modelling, STAT 4: LinReg(ax+b) fits a linear model, ExpReg fits y = a·bˣ ersions, the command can store it automatically) so you can graph the model against the scatter plot.
建模时，STAT 4: LinReg(ax+b) 拟合线性模型，ExpReg 拟合 y = a·bˣ，LnReg 拟合 y = a + b·ln x。务必指定列表，例如 LinReg(ax+b) L₁, L₂，并把回归方程复制到 Y=（较新系统可自动存储），以便把模型与散点图画在一起。
| Model | 模型 | Command | 命令 | Shape clue | 图像线索 |
| Linear | 线性 | LinReg(ax+b) | Constant first difference | 一阶差分恒定 |
| Exponential | 指数 | ExpReg | Constant ratio | 比值恒定 |
| Logarithmic | 对数 | LnReg | Rapid rise then flattening | 先快升后趋平 |
| Power | 幂函数 | PwrReg | Passes through origin, straight on log-log | 过原点，双对数图上为直线 |
Interpret r squared as the proportion of variation in the response explained by the model; a value of 0.94 means 94 percent of the variation is explained, not that the model is "94 percent correct".
把 r² 解释为模型所能解释的因变量变异比例；0.94 意味着解释了 94% 的变异，而不是"模型 94% 正确"。
9. Probability and Distributions | 概率与分布
All the distributions you need live under 2nd + DISTR (DISTR). normalcdf(lower, upper, μ, σ) gives the area under a normal curve between two bounds - use 1E99 for an open upper end. invNorm(area, μ, σ) is the inverse: give it a left-tail probability and it returns the boundary value. This single pair covers most AI HL normal distribution questions.
你需要的分布在 2nd + DISTR（DISTR）菜单里。normalcdf(lower, upper, μ, σ) 给出正态曲线在两点之间的面积——上端开放时用 1E99。invNorm(area, μ, σ) 是其反函数：给出左尾概率，返回边界值。这一对命令覆盖了大部分 AI HL 正态分布题目。
For discrete distributions, binompdf(n, p, k) gives exactly k successes and binomcdf(n, p, k) gives at most k successes. poissonpdf(λ, k) and poissoncdf(λ, k) work the same way for the Poisson distribution. Watch the wording: "at least 4" means 1 minus binomcdf(n, p, 3), and "between 3 and 7 inclusive" means binomcdf(n, p, 7) minus binomcdf(n, p, 2).
对离散分布，binompdf(n, p, k) 给出恰好 k 次成功，binomcdf(n, p, k) 给出至多 k 次成功。poissonpdf(λ, k) 与 poissoncdf(λ, k) 对泊松分布同理。注意措辞："at least 4" 等于 1 减 binomcdf(n, p, 3)，"between 3 and 7 inclusive" 等于 binomcdf(n, p, 7) 减 binomcdf(n, p, 2)。
Frequently confused pairs worth drilling:
值得反复练习的易混对：
- normalcdf vs invNorm: one goes from bounds to probability, the other from probability to a bound. | normalcdf 与 invNorm：一个由边界求概率，一个由概率求边界。
- pdf vs cdf: pdf is a single value, cdf is cumulative up to that value. | pdf 与 cdf：pdf 是单点值，cdf 是累积到该值的概率。
- μ and σ must be in the same units as the bounds you type. | μ 与 σ 必须与你输入的边界单位一致。
If a question pairs a normal model with a sample mean, remember that the standard error is σ ÷ √n, and the calculator does not know this automatically - you must enter it yourself.
若题目把正态模型与样本均值结合，记住标准误为 σ ÷ √n，计算器不会自动识别这一点，必须你自己输入。
10. Sequences, Series and Financial Mathematics | 数列、级数与金融数学
de and use L₁, L₂ with formulas entered in the STAT editor header, for example L₂ = 3·L₁ + 2, which fills a whole column at once.
AI HL 中的数列既有递推形式也有通项形式。要看项表，可切到 SEQ 模式输入 u(n) 与 u(nMin)；或留在 FUNCTION 模式，用 STAT 编辑器的列标题输入公式，例如 L₂ = 3·L₁ + 2，可一次填满整列。
To sum a series, use sum(seq(expression, variable, start, end)) from 2nd + LIST OPS. For example, the geometric sum of the first 20 terms with first term 5 and ratio 1.08 is sum(seq(5(1.08)n, n, 1, 20)). This removes the need to memorise the closed-form formula under exam pressure, though you should still be able to use it.
求级数和可用 2nd + LIST OPS 中的 sum(seq(表达式, 变量, 起, 止))。例如首项 5、公比 1.08 的等比数列前 20 项和为 sum(seq(5(1.08)n, n, 1, 20))。这在考试压力下免去记忆通项公式的需要，但你仍应能使用公式。
Financial mathematics is handled by APPS 1: Finance, 1: TVM Solver. The variables are N (number of payments), I% (annual nominal rate as a percentage), PV (present value), PMT (payment each period), FV (future value), P/Y and C/Y (payments and compounds per year). Enter every known value, set PMT: END for ordinary annuities and BEGIN for annuities due, then place the cursor on the unknown and press ALPHA + ENTER to solve.
金融数学由 APPS 1: Finance，1: TVM Solver 完成。变量包括 N（付款期数）、I%（年名义利率，以百分数输入）、PV（现值）、PMT（每期付款）、FV（终值）、P/Y 与 C/Y（每年付款次数与复利次数）。输入所有已知量，普通年金设 PMT: END，期初年金设 BEGIN，然后把游标放到未知量上按 ALPHA + ENTER 求解。
The cash-flow sign convention trips
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