PART I · SHORT RESPONSE第一部分 · 短答题SAT-style MCQ + ON/BC short answer · 22 marksSAT 风格选择题 + 安/卑省考短答 · 共 22 分
Section A · Short ResponseA 部分 · 短答题
Mix of multiple-choice and short-answer items. For MCQs, circle the letter; show enough work in the margin that a marker could verify. For short-answer items, state every domain restriction. No calculator on Q1–Q5.本节包含选择题与短答题。选择题请圈出字母答案，并在空白处写下足以让阅卷人核对的过程。短答题须写出每一个定义域限制条件。Q1–Q5 不可使用计算器。
Which of the following is the simplified form of $\dfrac{x^{2} - 25}{x^{2} - 4 x - 5}$, together with its restriction set?下列哪一项是 $\dfrac{x^{2} - 25}{x^{2} - 4 x - 5}$ 的最简形式（含定义域限制条件）？
(A) $\dfrac{x - 5}{x - 1}$, $x \ne 5$
(B) $\dfrac{x + 5}{x + 1}$, $x \ne -1, 5$
(C) $\dfrac{x + 5}{x - 1}$ (no restrictions)（无限制条件）
Combine each pair of rational expressions into a single rational expression in lowest terms. State all restrictions on the variable.将每一组有理表达式合并为一个最简形式的有理表达式，并写出变量的全部限制条件。
(a) $\dfrac{3}{x - 2} + \dfrac{5}{x + 4}$.。[3]
(b) $\dfrac{2 x}{x^{2} - 9} - \dfrac{1}{x - 3}$ (factor the first denominator before forming the LCD).（先因式分解第一个分母，再通分确定最小公分母）。[3]
Throughout this question, assume $x > 0$.本题始终假设 $x > 0$。
(a)Evaluate $27^{2/3}$ and $32^{-3/5}$ exactly, citing the identity $a^{m/n} = (\sqrt[n]{a})^{m}$.精确求出 $27^{2/3}$ 与 $32^{-3/5}$ 的值，并引用恒等式 $a^{m/n} = (\sqrt[n]{a})^{m}$。[2]
(b)Rewrite $\dfrac{\sqrt{x} \cdot \sqrt[3]{x^{2}}}{\sqrt[6]{x^{5}}}$ as a single power $x^{r}$ in lowest terms. Show the LCD-of-fractions step.将 $\dfrac{\sqrt{x} \cdot \sqrt[3]{x^{2}}}{\sqrt[6]{x^{5}}}$ 改写为最简的单一幂 $x^{r}$。请写出分数通分（最小公分母）的步骤。[3]
(c)Cite the Common Core standard HSN-RN.A.1 in one sentence: explain why the definition $a^{1/n} = \sqrt[n]{a}$ is forced by extending the exponent law $(a^{p})^{q} = a^{p q}$ from integer exponents to rational ones.引用共同核心标准 HSN-RN.A.1，用一句话解释：为何把指数律 $(a^{p})^{q} = a^{p q}$ 从整数指数延伸到有理指数后，定义 $a^{1/n} = \sqrt[n]{a}$ 是被迫成立的。[2]
PART II · EXTENDED RESPONSE第二部分 · 长答题AP-feeder FRQ + honors · 36 marksAP 衔接简答题 + 荣誉级 · 共 36 分
Section B · Extended ResponseB 部分 · 长答题
Show every algebraic step. State the domain restriction before clearing denominators (rational equations) or squaring (radical equations). For every candidate root, verify against the original equation and explicitly flag any extraneous solutions. No calculator on Q6–Q9.请写出每一步代数过程。在去分母（有理方程）或平方（根式方程）之前必须先写出定义域限制条件。对每一个候选根，须代回原方程验证，并明确标出任何增根（extraneous solution）。Q6–Q9 不可使用计算器。
(a)Factor every denominator and state the least common denominator (LCD). State the restriction set on $x$.将每一个分母因式分解，写出最小公分母（LCD），并写出 $x$ 的限制条件集合。[2]
(b)Combine $R(x)$ into a single rational expression over the LCD. Expand and collect the numerator, then factor it if possible.把 $R(x)$ 通分为最小公分母下的单一有理表达式。将分子展开并合并同类项，若可分解则再因式分解。[4]
(c)Cancel any common factor and state the simplified form of $R(x)$. Reaffirm the full restriction set (including any restriction that becomes invisible after cancellation, as required by HSA-APR.D.6).约去公因式后写出 $R(x)$ 的最简形式。须再次完整列出限制条件集合（包括约分后在表达式中"看不见"但仍存在的限制，依据 HSA-APR.D.6）。[2]
Solve each equation. In each part, state restrictions before clearing denominators, list every candidate root, and verify each candidate against the original equation. Explicitly flag any extraneous solutions.求解每一个方程。各小题须在去分母之前写出限制条件，列出全部候选根，并代回原方程逐一验证；如有增根（extraneous solution）须明确标出。
(c)In one sentence, cite Common Core HSA-REI.A.2 ("solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise") to explain why the verification step in part (a) was necessary even though the algebra was correct.用一句话引用共同核心 HSA-REI.A.2（"求解一元简单有理方程与根式方程，并举例说明增根如何产生"）解释：为何即使 (a) 的代数过程正确，验证步骤仍是必要的。[1]
Solve each radical equation by isolating, squaring, and checking against the original. For each part, state the radicand-non-negative condition before squaring, and flag any extraneous root.求解每一个根式方程：先将根号分离，再平方，然后代回原方程验证。每一小题须在平方之前写出根号下被开方数非负的条件，并标出任何多余根（增根）。
(a) $\sqrt{3 x + 1} = x - 1$ (single radical; one squaring).（单根号；平方一次即可）。[3]
(c)State the BC PC11 Big Idea phrasing the inquiry behind this question: "the meanings of, and connections between, operations extend to powers, radicals, and polynomials." Identify the one step in part (b) that introduces the possibility of an extraneous root.写出本题背后所对应的 BC PC11 大概念："运算的意义及其相互联系延伸至幂、根式与多项式"。指出 (b) 中引入增根可能性的那一步。[1]
Consider the rational function $f(x) = \dfrac{x^{2} - x - 6}{x^{2} - 4 x + 3}$. BC PC12's rational functions Content elaboration names the analysis below verbatim: asymptotes, intercepts, point discontinuities (holes), domain, end-behaviour.考虑有理函数 $f(x) = \dfrac{x^{2} - x - 6}{x^{2} - 4 x + 3}$。BC PC12 的有理函数内容细化逐项列出了以下分析：渐近线、截距、可去间断点（空洞）、定义域、末端行为。
(a)Factor both numerator and denominator. State any common factor.将分子和分母均因式分解，并写出公因式（若有）。[2]
(b)Identify the point discontinuity (hole) of $f$. Give the coordinates $(a, f^{*}(a))$ where $f^{*}$ is the function after cancellation.找出 $f$ 的可去间断点（空洞）。给出坐标 $(a, f^{*}(a))$，其中 $f^{*}$ 是约去公因式后的函数。[2]
(c)Identify any vertical asymptote(s). Justify by citing the rule "denominator-only zero $\Rightarrow$ VA; common-factor zero $\Rightarrow$ hole."找出所有垂直渐近线。请引用规则"仅分母为零 $\Rightarrow$ 垂直渐近线；公因式为零 $\Rightarrow$ 空洞"加以说明。[2]
(d)Determine the horizontal asymptote of $f$ by degree comparison. Then state the $x$-intercept(s) and the $y$-intercept of $f$.用次数比较法确定 $f$ 的水平渐近线，并写出 $f$ 的 $x$ 截距与 $y$ 截距。[3]
(e)State the domain of $f$ in set-builder notation. List every value of $x$ excluded by the original denominator (the cancelled hole is still excluded, as the BC PC12 "domain" elaboration requires).用集合构造记号写出 $f$ 的定义域。请列出原分母排除的每一个 $x$ 值（按 BC PC12 定义域细化的要求，被约去的空洞仍需排除）。[2]
PART III · MODELING / APPLIED第三部分 · 建模与应用Universal · 32 marks通用 · 共 32 分
Section C · Modeling and ApplicationsC 部分 · 建模与应用
Name your variables (with units) before writing equations. State the situational domain. Set up your model, solve, and conclude with a one-sentence answer in context. Calculator permitted throughout Part III.写出方程之前请先命名变量并标明单位，并写出情境（实际）定义域。建立模型、求解，最后用一句话结合情境作答。第三部分全程可使用计算器。
The period $T$ (in seconds) of a simple pendulum of length $L$ (in metres) is modeled by $T = 2 \pi \sqrt{L / g}$, where $g = 9.8$ m/s$^{2}$ is the acceleration due to gravity near Earth's surface.长度为 $L$（米）的单摆，其周期 $T$（秒）由 $T = 2 \pi \sqrt{L / g}$ 给出，其中 $g = 9.8$ m/s$^{2}$ 是地球表面附近的重力加速度。
(a)A grandfather clock has a pendulum of length $L = 0.994$ m. Compute its period $T$ to three decimal places. Show the substitution into the radical model.一座落地老式时钟的摆长 $L = 0.994$ 米。求其周期 $T$，保留三位小数，并写出代入根式模型的过程。[2]
(b)Solve the model for $L$ in terms of $T$. (Square both sides; isolate $L$.) State the situational domain $L > 0$, $T > 0$.由模型解出 $L$ 关于 $T$ 的表达式（两边平方，然后分离 $L$）。写出情境定义域 $L > 0$、$T > 0$。[3]
(c)A playground swing is designed to have period $T = 3.5$ s. Use part (b) to determine the required length $L$, rounded to two decimal places. State units.游乐场秋千的设计周期为 $T = 3.5$ 秒。用 (b) 的结果确定所需长度 $L$，保留两位小数，并写明单位。[3]
(d)Verify your answer to part (c) by substituting back into the original model $T = 2 \pi \sqrt{L / g}$. The verification step is required by Common Core HSA-REI.A.2 for radical equations , cite the standard and explain why the verification is structurally needed (squaring is a non-injective operation).将 (c) 的答案代回原模型 $T = 2 \pi \sqrt{L / g}$ 进行验证。共同核心 HSA-REI.A.2 要求根式方程必须做这一验证步骤——请引用该标准，并说明为何验证在结构上是必需的（因为平方运算并非单射）。[2]
Two students paint a fence. Working alone, Aanya finishes the job in $x$ hours; Ben finishes it in $x + 3$ hours. Together they finish in $2$ hours.两位学生粉刷一段围栏。单独工作时，Aanya 完成任务需要 $x$ 小时；Ben 需要 $x + 3$ 小时。两人合作 $2$ 小时即可完成。
(a)State Aanya's rate of work and Ben's rate of work, each as a fraction of the job per hour, using $x$.用 $x$ 表示 Aanya 与 Ben 的工作效率，分别写成每小时完成的工作量（分数形式）。[2]
(b)Write a rational equation in $x$ that expresses the combined-rate condition "in $2$ hours together, they complete $1$ full job." State the restriction $x > 0$ and any other restriction implied by the algebra.写出关于 $x$ 的一个有理方程，表达"两人合作 $2$ 小时完成 $1$ 件完整工作"的合作效率条件。写出限制条件 $x > 0$ 以及代数运算所要求的其他限制条件。[2]
(c)Multiply through by the LCD and reduce the equation to a quadratic in $x$. Solve the quadratic exactly using the quadratic formula from Unit 2.两边同乘最小公分母，将方程化为关于 $x$ 的二次方程。使用第 2 单元中的求根公式精确求解该二次方程。[4]
(d)One root is positive, one is negative. Reject the negative root with a one-sentence justification (the situational domain $x > 0$ comes from the modeling context, not just the algebraic restrictions). State Aanya's time and Ben's time, each rounded to one decimal place.两个根一正一负。用一句话说明为何舍去负根（情境定义域 $x > 0$ 来自建模背景，并不仅仅来自代数限制条件）。分别写出 Aanya 与 Ben 的所用时间，保留一位小数。[2]
(e)Verify your answer by computing $1/x + 1/(x + 3)$ at the value found in (d) and checking that the sum equals $1/2$ (the combined hourly rate). Round to three decimal places.在 (d) 得到的 $x$ 值处计算 $1/x + 1/(x + 3)$，验证其和等于 $1/2$（合作每小时的效率）。保留三位小数。[1]
Q12HARD难🇨🇦 BC卑🇺🇸 US美BC Provincial-style卑诗省考风格§2 + §4 Rational Function App有理函数应用 · BC PC12 / HSA-CED.A.1[11 marks][11 分]
A small bottling plant has fixed daily costs of CA$240 and a variable cost of CA$0.60 per bottle. The plant produces $x$ bottles per day ($x > 0$). The average cost per bottle, in dollars, is the rational function $\bar C(x) = \dfrac{240 + 0.60 x}{x}$.一家小型瓶装厂每日固定成本为加币 CA$240，每瓶可变成本为 CA$0.60。工厂每日生产 $x$ 瓶（$x > 0$）。每瓶的平均成本（以加元计）为有理函数 $\bar C(x) = \dfrac{240 + 0.60 x}{x}$。
(a)Rewrite $\bar C(x)$ as a sum: $\bar C(x) = \dfrac{240}{x} + 0.60$. Identify which term came from fixed cost and which from variable cost. (This is the §2 "split a single rational into a sum" move in reverse.)将 $\bar C(x)$ 写成和的形式：$\bar C(x) = \dfrac{240}{x} + 0.60$。指出哪一项来自固定成本，哪一项来自可变成本。（这是 §2"把单一有理表达式拆分为求和"的逆向操作。）[2]
(b)Compute $\bar C(100)$, $\bar C(400)$, and $\bar C(1200)$. Round each to the nearest cent. Briefly interpret why $\bar C$ decreases as $x$ increases.计算 $\bar C(100)$、$\bar C(400)$ 与 $\bar C(1200)$，各四舍五入到分（两位小数）。简要解释为何 $\bar C$ 随 $x$ 增加而减小。[3]
(c)Determine the horizontal asymptote of $\bar C(x)$ by degree comparison, and interpret its meaning as $x \to \infty$ in context (the asymptotic per-bottle cost a high-volume plant approaches but never reaches).用次数比较法确定 $\bar C(x)$ 的水平渐近线，并结合情境解释当 $x \to \infty$ 时的含义（即大产量工厂趋近但永远达不到的每瓶渐近成本）。[3]
(d)The plant is profitable if and only if $\bar C(x) \le 0.85$ (the wholesale price per bottle). Set up a rational equation $\bar C(x) = 0.85$, clear the denominator, and solve for the break-even production $x$.当且仅当 $\bar C(x) \le 0.85$（每瓶批发价）时工厂盈利。列出有理方程 $\bar C(x) = 0.85$，去分母后求出盈亏平衡时的产量 $x$。[2]
(e)State the situational domain of $\bar C$. Cite the BC PC12 rational functions elaboration (asymptotes, intercepts, point discontinuities, domain, end-behaviour) to justify that $x = 0$ is excluded both algebraically (denominator-only zero $\Rightarrow$ vertical asymptote) and contextually (a plant producing zero bottles has undefined per-bottle cost).写出 $\bar C$ 的情境定义域。引用 BC PC12 有理函数细化（渐近线、截距、可去间断点、定义域、末端行为）说明为何 $x = 0$ 在代数上（仅分母为零 $\Rightarrow$ 垂直渐近线）和情境上（生产 $0$ 瓶时每瓶成本无定义）都应当排除。[1]
🇺🇸 US Common Core美国共同核心HSA-APR.D.6 · HSA-APR.D.7 (+) · HSA-REI.A.2 · HSA-REI.D.11 · HSA-CED.A.1 · HSN-RN.A.1 · HSN-RN.A.2 · HSF-IF.B.4
🇨🇦 British Columbia不列颠哥伦比亚PC11 Content rational expressions and equations, radical operations and equations, powers with rational exponents · PC11 Big Idea "meanings of operations extend to powers, radicals, and polynomials" · PC12 Content rational functions (asymptotes, intercepts, point discontinuities, domain, end-behaviour)PC11 内容 有理表达式与方程、根式运算与方程、有理指数的幂 · PC11 大概念 "运算的意义延伸至幂、根式与多项式" · PC12 内容 有理函数（渐近线、截距、可去间断点、定义域、末端行为）
Full 3-column Syllabus Map lives in ../Study Guides/Unit_4_Rational_and_Radical_Expressions.html.完整的三列大纲对照表见 ../Study Guides/Unit_4_Rational_and_Radical_Expressions.html。