PART I · SHORT RESPONSE第一部分 · 短答题SAT-style MCQ + ON/BC short answer · 18 marksSAT 风格选择题 + 安/卑省考短答 · 共 18 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, name the parameter you are reading ($a$, $b$, $h$, or $k$) before you commit. 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。短答题在动笔前先点名所读的参数（$a$、$b$、$h$ 或 $k$）。Q1–Q5 不可使用计算器。
The point $(-3, 5)$ lies on the graph of $y = f(x)$. What is the corresponding point on the graph of $y = f(-x)$?点 $(-3, 5)$ 在 $y = f(x)$ 的图像上。该点在 $y = f(-x)$ 图像上的对应点是？
The point $(8, 3)$ lies on the graph of $y = f(x)$.点 $(8, 3)$ 在 $y = f(x)$ 的图像上。
(a)State the corresponding point on the graph of $y = f(4 x)$, and identify whether this is a horizontal stretch or a horizontal compression.写出该点在 $y = f(4 x)$ 图像上的对应点，并判别这是水平伸还是水平压缩。[2]
(b)State the corresponding point on the graph of $y = \tfrac{1}{2} f(x)$, and identify the vertical transformation.写出该点在 $y = \tfrac{1}{2} f(x)$ 图像上的对应点，并指明竖直方向的变换。[2]
Q5MEDIUM中🇨🇦 BC卑🇨🇦 AB艾BC Provincial-style卑诗省考风格§1 + §3 Coordinate Rules坐标变换规则 · BC PC 12 / AB Math 30-1 RF 2.1, 5.1[5 marks][5 分]
The graph of $y = \sqrt{x}$ is transformed in three independent ways. For each transformed graph, state the image equation and give the image of the parent-graph point $(4, 2)$.$y = \sqrt{x}$ 的图像分别进行三次独立的变换。对每个变换后的图像，写出其方程，并给出母函数图像上点 $(4, 2)$ 的像。
(a)Shift the graph $3$ units to the right and $1$ unit down.将图像向右平移 $3$ 个单位、向下平移 $1$ 个单位。[2]
(b)Reflect the graph over the $x$-axis.将图像关于 $x$ 轴反射。[2]
(c)Reflect the graph over the $y$-axis (note: the resulting graph is no longer the principal $\sqrt{x}$).将图像关于 $y$ 轴反射（注意：所得图像已不再是主值 $\sqrt{x}$）。[1]
PART II · EXTENDED RESPONSE第二部分 · 简答题AP-feeder FRQ + honors · 35 marksAP 衔接简答题 + 荣誉级 · 共 35 分
Section B · Extended ResponseB 部分 · 简答题
Show every algebraic step. For master-form reads, factor the $b$ out of the bracket before you state $h$. For composition, write the inner-then-outer substitution explicitly. For inverses, use the swap-and-solve method and state any domain restriction. No calculator on Q6–Q9 unless noted.每一步代数运算都要写出。读取一般形式时，请在写出 $h$ 之前先把 $b$ 提到括号外。对函数复合，要明确写出"先内后外"的代入过程。对反函数，使用"互换求解"法（swap-and-solve）并写出任何定义域的限制条件。除特别说明外，Q6–Q9 不可使用计算器。
Consider the function $y = -2 f\bigl(\tfrac{1}{3}(x - 6)\bigr) + 4$, where $f$ is a given parent function.考虑函数 $y = -2 f\bigl(\tfrac{1}{3}(x - 6)\bigr) + 4$，其中 $f$ 是给定的母函数。
(a)Identify the four master-form parameters $a$, $b$, $h$, $k$.写出一般形式中的四个参数 $a$、$b$、$h$、$k$。[2]
(b)Describe in words the four geometric transformations applied to $y = f(x)$, in the correct order of operations (horizontal first, then vertical).用文字描述施加于 $y = f(x)$ 的四种几何变换，并按正确的操作顺序排列（先水平后竖直）。[4]
(c)The point $(3, 1)$ lies on $y = f(x)$. Find its image on the transformed graph.点 $(3, 1)$ 在 $y = f(x)$ 的图像上。求该点在变换后图像上的像。[2]
Consider the transformed function $y = f(2 x - 10) + 3$, where $f$ is a parent function.考虑变换后的函数 $y = f(2 x - 10) + 3$，其中 $f$ 为母函数。
(a)Rewrite $y = f(2 x - 10) + 3$ in master form $y = a f(b(x - h)) + k$ by factoring the $b$ out of the bracket. State $a$, $b$, $h$, $k$.通过将 $b$ 提到括号外，把 $y = f(2 x - 10) + 3$ 改写为一般形式 $y = a f(b(x - h)) + k$。写出 $a$、$b$、$h$、$k$。[3]
(b)List the four transformations applied to $y = f(x)$, in order of operations. Be explicit about whether the horizontal compression precedes or follows the horizontal shift, and explain why.按操作顺序列出施加于 $y = f(x)$ 的四种变换。明确说明水平压缩是发生在水平平移之前还是之后，并解释原因。[3]
(c)The parent graph passes through $(0, 0)$, $(2, 4)$, and $(4, 0)$. Find the images of these three points on the transformed graph.母函数图像过 $(0, 0)$、$(2, 4)$ 和 $(4, 0)$。求这三点在变换后图像上的像。[3]
Q8MEDIUM中🇨🇦 BC卑🇨🇦 AB艾BC Provincial-style卑诗省考风格§7 Inverses (swap-and-solve)反函数（互换求解法） · AB Math 30-1 RF 6.1–6.5[9 marks][9 分]
Let $f(x) = 3 x - 5$ and $g(x) = (x - 1)^{2} + 2$.设 $f(x) = 3 x - 5$，$g(x) = (x - 1)^{2} + 2$。
(a)Determine $f^{-1}(x)$ using the swap-and-solve method. State its domain and range.用互换求解法（swap-and-solve）求 $f^{-1}(x)$，并写出其定义域与值域。[3]
(b)Verify algebraically that $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$.用代数方法验证：对 $f^{-1}$ 定义域内的所有 $x$，都有 $f(f^{-1}(x)) = x$。[2]
(c)Explain why $g$ does not have an inverse on its natural domain. State a domain restriction for $g$ that produces a one-to-one function, and find $g^{-1}(x)$ on that restricted domain.解释为何 $g$ 在其自然定义域上不存在反函数。给出使 $g$ 成为一一对应函数的一个定义域限制，并在该受限定义域上求 $g^{-1}(x)$。[4]
(a)Compute $(f \circ g)(3)$ and $(g \circ f)(10)$. Show the inner-then-outer substitution explicitly.计算 $(f \circ g)(3)$ 与 $(g \circ f)(10)$，并明确写出"先内后外"的代入过程。[2]
(b)Find an explicit formula for $(f \circ g)(x)$, and state its domain. (Hint: the domain restriction is "the inner output must be a legal input to the outer".)写出 $(f \circ g)(x)$ 的显式表达式，并说明其定义域。（提示：定义域的限制条件是"内函数的输出必须是外函数允许的输入"。）[3]
(c)Find an explicit formula for $(g \circ f)(x)$, and state its domain.写出 $(g \circ f)(x)$ 的显式表达式，并说明其定义域。[2]
(d)Demonstrate by counter-example that composition is not commutative: produce an input $x$ for which $(f \circ g)(x) \ne (g \circ f)(x)$.举反例说明函数复合不满足交换律：给出一个 $x$，使 $(f \circ g)(x) \ne (g \circ f)(x)$。[2]
PART III · MODELING / APPLIED第三部分 · 建模与应用Universal · 28 marks通用题型 · 共 28 分
Section C · Modeling and ApplicationsC 部分 · 建模与应用
Name your variables (with units) before writing equations. Set up your model, solve, and conclude with a one-sentence answer in context. Calculator permitted throughout Part III.在写方程前，先定义变量名（含单位）。建立模型、求解，并以一句结合情境的话作答。第三部分全程可用计算器。
Q10MEDIUM中🇺🇸 US美AP-feeder FRQAP 衔接简答题§4 Master Form on $\sqrt{x}$$\sqrt{x}$ 上的一般形式 · HSF-BF.B.3[9 marks][9 分]
Let $f(x) = \sqrt{x}$ (the parent square-root function), and define $T(x) = 2 \sqrt{x - 3} + 1$.设 $f(x) = \sqrt{x}$（母平方根函数），并定义 $T(x) = 2 \sqrt{x - 3} + 1$。
(a)Express $T(x)$ in master form $y = a f(b(x - h)) + k$, and state $a$, $b$, $h$, $k$.将 $T(x)$ 写成一般形式 $y = a f(b(x - h)) + k$，并写出 $a$、$b$、$h$、$k$。[2]
(b)List the geometric transformations applied to $f$, in order of operations.按操作顺序列出施加于 $f$ 的几何变换。[2]
(c)State the domain and the range of $T$ in interval notation. Justify your answer by tracking the parent's domain and range through each transformation.用区间记号写出 $T$ 的定义域与值域。逐步追踪母函数的定义域与值域经过每次变换后的变化，以此为答案提供依据。[3]
(d)The parent function passes through $(0, 0)$, $(1, 1)$, and $(4, 2)$. Find the images of these three points on the graph of $T$.母函数图像过 $(0, 0)$、$(1, 1)$ 与 $(4, 2)$。求这三点在 $T$ 的图像上的像。[2]
A Canadian ski resort posts temperature in degrees Celsius and altitude in metres. Let $C(a) = 12 - 0.0065 \, a$ be the air temperature in °C at altitude $a$ metres above the base (a standard atmospheric lapse-rate model). A US visitor wants the temperature in °F. The Fahrenheit conversion is $F(c) = \tfrac{9}{5} c + 32$.某加拿大滑雪场以摄氏度公布气温，以米记录海拔。设 $C(a) = 12 - 0.0065 \, a$ 为山脚之上 $a$ 米处的气温（°C），这是标准大气递减率模型。一位美国游客想看华氏度。华氏度换算公式为 $F(c) = \tfrac{9}{5} c + 32$。
(a)Write a composite function $H(a) = (F \circ C)(a)$ giving the temperature in °F as a function of altitude in metres. Simplify to a linear expression in $a$.写出复合函数 $H(a) = (F \circ C)(a)$，用以表示华氏温度关于海拔（米）的函数。化简为关于 $a$ 的线性表达式。[3]
(b)Use $H(a)$ to find the Fahrenheit temperature at altitude $a = 2000$ m.用 $H(a)$ 求海拔 $a = 2000$ 米处的华氏温度。[2]
(c)Find $H^{-1}(F)$ using the swap-and-solve method; state what its output represents in context.用互换求解法求 $H^{-1}(F)$；说明其输出在情境中代表什么。[3]
(d)Use $H^{-1}$ to find the altitude (to the nearest metre) at which the air temperature is $32^{\circ}\mathrm{F}$. Interpret in one sentence.用 $H^{-1}$ 求气温为 $32^{\circ}\mathrm{F}$ 时的海拔（精确到米）。用一句话作情境解释。[1]
Q12HARD难Honors荣誉级🇨🇦 BC卑🇺🇸 US美BC Provincial-style卑诗省考风格§4 + §7 Synthesis on a Cubic三次函数综合 · BC PC 12 / HSF-BF.B.4[10 marks][10 分]
Let $f(x) = x^{3}$ (the parent cubic), and define $T(x) = -2(x + 1)^{3} + 4$.设 $f(x) = x^{3}$（母三次函数），并定义 $T(x) = -2(x + 1)^{3} + 4$。
(a)Express $T(x)$ in master form $y = a f(b(x - h)) + k$, and state $a$, $b$, $h$, $k$.将 $T(x)$ 写成一般形式 $y = a f(b(x - h)) + k$，并写出 $a$、$b$、$h$、$k$。[2]
(b)Describe the four geometric transformations applied to $f$, in order.按顺序描述施加于 $f$ 的四种几何变换。[2]
(c)The parent cubic $f(x) = x^{3}$ is a one-to-one odd function on its full domain. Explain in one or two sentences why $T$ remains one-to-one (so $T^{-1}$ exists on all of $\mathbb{R}$).母三次函数 $f(x) = x^{3}$ 在整个定义域上是一一对应的奇函数。用一两句话解释为何 $T$ 仍保持一一对应（即 $T^{-1}$ 在整个 $\mathbb{R}$ 上存在）。[1]
(d)Find $T^{-1}(x)$ using the swap-and-solve method. State the domain and the range of $T^{-1}$.用互换求解法求 $T^{-1}(x)$，并写出 $T^{-1}$ 的定义域与值域。[4]
(e)Verify your inverse by evaluating $T(T^{-1}(4))$.通过计算 $T(T^{-1}(4))$ 验证所求的反函数。[1]
🇺🇸 US Common Core美国共同核心HSF-BF.A.1 · HSF-BF.B.3 · HSF-BF.B.4 · HSF-BF.A.1.c (+)for composition用于函数复合
🇨🇦 Ontario / BC安大略 / 卑诗MCR3U A1.4–A1.9 (transformations + inverses of linear/quadratic) · MHF4U Strand C (composition) · BC PC 12 Transformations (translations, stretches, reflections, inverses) + composed functions extensionMCR3U A1.4–A1.9（一次 / 二次函数的变换与反函数） · MHF4U 单元 C（函数复合） · BC PC 12 变换（平移、伸缩、反射、反函数）+ 复合函数拓展
Full 4-column Syllabus Map lives in ../Study Guides/Unit_10_Function_Transformations_and_Composition.html.完整的四列大纲对照表见 ../Study Guides/Unit_10_Function_Transformations_and_Composition.html。