📚 Chapter Review 4: Graphs and Transformations | 第4章复习：函数图像与变换
eciprocal graphs, and transforming the graph of any given function y = f(x). The focus is on efficient curve sketching and on writing the correct new equation after translations, stretches and reflections.
本章复习巩固爱德思 A-Level 纯数学第 4 章的核心技能：识别并绘制三次、四次和倒数函数图像，以及对任意给定函数 y = f(x) 进行图像变换。重点是高效绘制曲线，以及在平移、伸缩和反射之后写出正确的新方程。
1. Sketching Cubic Graphs | 绘制三次函数图像
0, the curve falls to −∞ on the left and rises to +∞ on the right; if a < 0, this behaviour is reversed.
三次曲线的一般式为 y = ax³ + bx² + cx + d，其中 a ≠ 0。首项系数 a 的正负决定两端走势：若 a > 0，曲线左侧趋向 −∞、右侧趋向 +∞；若 a < 0 则相反。
gives the point (0, d). Then factorise where possible to locate the x-intercepts, and use the multiplicity of each root to decide how the curve behaves at the axis.
绘图时先令 x = 0 求出 y 轴截距，即点 (0, d)；再尽可能因式分解以确定 x 轴截距，并根据每个根的重数判断曲线在 x 轴处的走向。
A repeated factor gives a stationary point on the x-axis, while a triple factor gives an inflection point that crosses the axis with zero gradient. A single factor always gives a simple crossing.
重复因式表示 x 轴处有驻点（曲线相切并转向）；三重因式表示曲线以零斜率穿过 x 轴，此处为拐点；单因式则表示曲线简单地穿过 x 轴。
x-axis and turns. Expanding gives y = x³ + 4x² − 3x − 18, so the y-intercept is (0, −18).
例如，y = (x − 2)(x + 3)² 在 x = 2 处有一个单根，曲线穿过 x 轴；在 x = −3 处有重根，曲线在 x 轴处相切并转向。展开得 y = x³ + 4x² − 3x − 18，因此 y 轴截距为 (0, −18)。
2. Sketching Quartic Graphs | 绘制四次函数图像
es a W-shape or a U-shape. If a > 0, both ends rise to +∞; if a < 0, both ends fall to −∞.
四次函数图像的一般式为 y = ax⁴ + bx³ + cx² + dx + e，其中 a ≠ 0。根据重根的不同，图像呈 W 形或 U 形。若 a > 0，两端都上升至 +∞；若 a < 0，两端都下降至 −∞。
Repeated factors such as (x − p)² create a bounce at the x-axis, while a single factor creates a crossing. Because a quartic has at most four linear factors, the shape is determined by how those factors repeat.
诸如 (x − p)² 的重复因式使曲线在 x 轴处"弹跳"（相切后返回同侧），单因式则使曲线穿过 x 轴。由于四次式最多有四个一次因式，图像形状完全由这些因式的重复方式决定。
tion-style touch that crosses the axis with zero gradient, while the remaining linear factor gives an ordinary crossing.
四次式中的三重因式，例如 y = (x + 1)³(x − 3)，会在 x 轴处产生一个零斜率穿过的拐点式接触，而剩下的一次因式则给出普通的穿轴点。
touches and turns at x = −1 and at x = 2, crosses nowhere, and its y-intercept is (0, 4).
例如：y = (x + 1)²(x − 2)² 为 W 形，a > 0。曲线在 x = −1 和 x = 2 处相切并转向，不与 x 轴相交，y 轴截距为 (0, 4)。
3. Reciprocal Graphs and Asymptotes | 倒数函数图像与渐近线
h is defined at x = 0, so both have a vertical asymptote at x = 0, drawn as a dashed line.
必须掌握的两类倒数函数图像是 y = a/x 与 y = a/x²，其中 a 为非零常数。两者在 x = 0 处都无定义，因此都有竖直渐近线 x = 0，绘图时用虚线表示。
also has a horizontal asymptote at y = 0, and the curve is symmetric about the line y = x.
当 a > 0 时，y = a/x 的图像有两支：一支在第一象限，一支在第三象限，都无限接近坐标轴。它还有水平渐近线 y = 0，并且曲线关于直线 y = x 对称。
For y = a/x² with a > 0, both branches lie above the x-axis — is graph is symmetric about the y-axis. If a < 0, every branch is reflected in the x-axis.
当 a > 0 时，y = a/x² 的两支都位于 x 轴上方，分居 y 轴两侧。当 x → 0 时 y → +∞；当 x → ±∞ 时 y → 0。该图像关于 y 轴对称。若 a < 0，则各支关于 x 轴反射。
l both asymptotes and mark one or two key points, such as (1, a) and (−1, a) for y = a/x².
绘制倒数函数图像时，务必标注两条渐近线，并标出一两个关键点，例如 y = a/x² 上的 (1, a) 与 (−1, a)。
4. A Reliable Sketching Checklist | 一套可靠的绘图检查清单
Efficient sketching in the exam means working through the same short checklist every time, rather than plotting many individual points. Examiners reward correct shape, correct intercepts and correct asymptotic behaviour.
考试中高效绘图意味着每次都按同一份简短清单操作，而不是逐点描图。评分看重的是形状正确、截距正确以及渐近行为正确。
- Shape and end behaviour: decide the sign of the leading coefficient and how the ends behave.
- y-intercept: substitute x = 0.
- x-intercepts: factorise and record the multiplicity of each root.
- Turning or touching points: mark where the curve touches and turns.
- Asymptotes: draw every asymptote as a dashed line and label its equation.
- 形状与两端走势：确定首项系数的正负以及两端的走向。
- y 轴截距：代入 x = 0。
- x 轴截距：因式分解并记录每个根的重数。
- 转向或相切点：标出曲线相切并转向的位置。
- 渐近线：每条渐近线都用虚线画出并标出方程。
A clear sketch is a communication tool: reviewers should be able to read off all the key features without any additional explanation.
清晰的图像是一种表达工具：阅卷人应当能直接从图中读出所有关键特征，无需额外解释。
5. Function Notation and Transformations | 函数记号与图像变换
x), then every transformation produces a new graph whose equation can be written using f again.
本章所有图像变换都用函数 f 来描述。若已知 y = f(x) 的图像，那么每一种变换都能得到一条新曲线，其方程仍可用 f 表示。
), then the transformed graph contains the image of that point, and the relationship between the old and the new coordinates tells you the type of transformation.
核心思想是"一般点的映射"。若点 (x, y) 在 y = f(x) 上，则变换后的图像包含该点的像；新旧坐标之间的关系就决定了变换的类型。
There are three families to keep separate: translations (the shape is unchanged, the curve slides), stretches (the shape is distorted in one direction only) and reflections (the shape is flipped in a line).
需要区分三大类：平移（形状不变，曲线滑动）、伸缩（只在某一个方向上拉伸或压缩形状）以及反射（形状关于某条线翻转）。
6. Translations | 平移变换
x) + a is the graph of y = f(x) moved a units upwards, with a < 0 giving a downwards move.
在函数整体上加常数会使图像竖直平移。y = f(x) + a 的图像即 y = f(x) 向上移动 a 个单位；若 a < 0 则向下移动。
y = f(x) + a : (x, y) → (x, y + a)
e opposite of what many students expect: y = f(x + a) moves the curve a units to the left.
把 x 换成 x + a 会使图像水平平移，而且方向常常与直觉相反：y = f(x + a) 是把曲线向左移动 a 个单位。
y = f(x + a) : (x, y) → (x − a, y) , shift of a units left
at x = c becomes a vertical asymptote at x = c − a under the transformation y = f(x + a).
平移不改变形状、方向，也不改变任何水平或竖直线段的长度。在变换 y = f(x + a) 下，竖直渐近线 x = c 变为 x = c − a。
7. Stretches | 伸缩变换
Multiplying the whole function by a constant stretches the graph vertically with scale factor a. Every y-coordinate is multiplied by a, while x-coordinates stay the same.
函数整体乘以常数 a 会使图像沿竖直方向以比例因子 a 伸缩。所有 y 坐标乘以 a，而 x 坐标保持不变。
y = a f(x) : (x, y) → (x, ay)
Replacing x by ax stretches the graph horizontally, and once again the factor appears "reversed": y = f(ax) is a horizontal stretch with scale factor 1/a about the y-axis.
把 x 换成 ax 会使图像沿水平方向伸缩，同样地，比例因子是"倒数"形式：y = f(ax) 是关于 y 轴、比例因子为 1/a 的水平伸缩。
y = f(ax) : (x, y) → (x ÷ a, y)
ept at x = p ÷ a under y = f(ax). A y-intercept at (0, q) becomes (0, aq) under y = af(x).
伸缩会改变形状。根的位置也会移动：y = f(x) 在 x = p 处的 x 轴截距，在 y = f(ax) 下变为 x = p ÷ a；y 轴截距 (0, q) 在 y = af(x) 下变为 (0, aq)。
8. Reflections | 反射变换
s the graph in the x-axis. Every point (x, y) maps to (x, −y), so a maximum becomes a minimum and the curve is turned upside down.
变换 y = −f(x) 表示图像关于 x 轴反射。每一点 (x, y) 映射到 (x, −y)，极大值变为极小值，曲线整体上下翻转。
y = −f(x) : (x, y) → (x, −y)
so roots change sign and a curve with a vertical asymptote at x = c now has one at x = −c.
变换 y = f(−x) 表示图像关于 y 轴反射。每一点 (x, y) 映射到 (−x, y)，因此根改变符号，原本在 x = c 有竖直渐近线的曲线，现在渐近线在 x = −c。
y = f(−x) : (x, y) → (−x, y)
hose right end rises to +∞, once reflected in the x-axis, has its right end falling to −∞.
反射不改变图像的大小，只改变方向。a > 0、右端上升至 +∞ 的三次曲线，关于 x 轴反射后右端便下降至 −∞。
9. Combining Transformations and Writing New Equations | 变换的组合与新方程书写
Exam questions often apply two transformations to the same curve. The safest method is to apply them in the order stated, and to work with function notation until the final step.
考试题常在同一曲线上施加两种变换。最稳妥的方法是按题目给出的顺序依次施加，并在最后一步之前一直使用函数记号。
ating 3 units right, stretching vertically with scale factor 2, and translating 1 unit up.
例如，给定 y = f(x)，曲线 y = 2f(x − 3) + 1 可这样得到：先向右平移 3 个单位，再沿竖直方向以比例因子 2 伸缩，最后向上平移 1 个单位。
efully. If f(x) = x³ − 4x, then f(x + 1) − 2 = (x + 1)³ − 4(x + 1) − 2 = x³ + 3x² − x − 5.
代入多项式时要仔细展开。若 f(x) = x³ − 4x，则 f(x + 1) − 2 = (x + 1)³ − 4(x + 1) − 2 = x³ + 3x² − x − 5。
A useful summary table of mappings for a general point (x, y) is given below.
下表汇总了一般点 (x, y) 在各类变换下的映射关系。
| Transformation | 变换 | Equation | 方程 | Mapping | 映射 |
| Vertical translation | 竖直平移 | y = f(x) + a | (x, y) → (x, y + a) |
| Horizontal translation | 水平平移 | y = f(x + a) | (x, y) → (x − a, y) |
| Vertical stretch | 竖直伸缩 | y = a f(x) | (x, y) → (x, ay) |
| Horizontal stretch | 水平伸缩 | y = f(ax) | (x, y) → (x ÷ a, y) |
| Reflection in x-axis | 关于 x 轴反射 | y = −f(x) | (x, y) → (x, −y) |
| Reflection in y-axis | 关于 y 轴反射 | y = f(−x) | (x, y) → (−x, y) |
10. Transforming Key Points and Asymptotes | 关键点与渐近线的变换
When a curve has asymptotes, they must be transformed along with the rest of the graph. Forgetting this is one of the most common sources of lost marks.
当曲线带有渐近线时，渐近线必须与图像其余部分一起变换。忘记这一点是最常见的失分原因之一。
to x = c − a. Under y = f(ax), it moves to x = c ÷ a. Under y = f(−x), it moves to x = −c.
在 y = f(x + a) 下，竖直渐近线 x = c 移到 x = c − a；在 y = f(ax) 下移到 x = c ÷ a；在 y = f(−x) 下移到 x = −c。
mptote y = k of y = f(x) becomes y = k + a under y = f(x) + a, and y = ak under y = af(x).
水平渐近线的变化规律与 y 坐标相同。y = f(x) 的水平渐近线 y = k，在 y = f(x) + a 下变为 y = k + a，在 y = af(x) 下变为 y = ak。
ptotes x = 2 and y = 3, and is the original curve translated 2 units right and 3 units up.
例如：曲线 y = 1/x 的渐近线为 x = 0 与 y = 0。曲线 y = 1 ÷ (x − 2) + 3 的渐近线为 x = 2 与 y = 3，它是原曲线向右平移 2 个单位、向上平移 3 个单位得到的。
11. Worked Exam-Style Example | 考试型例题精讲
d the equation of the curve after each transformation, and state how the coordinates of a general point change.
曲线 C 的方程为 y = x³ − 4x。求下列各变换后曲线的方程，并说明一般点坐标如何变化。
s y = f(x + 1) − 2, so the new equation is y = (x + 1)³ − 4(x + 1) − 2 = x³ + 3x² − x − 5.
（a）向左平移 1 个单位、向下平移 2 个单位。即 y = f(x + 1) − 2，故新方程为 y = (x + 1)³ − 4(x + 1) − 2 = x³ + 3x² − x − 5。
r 2, followed by a reflection in the x-axis. This is y = −2f(x) = −2(x³ − 4x) = −2x³ + 8x.
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