📚 IB Mathematics: Graphing Functions | IB数学：函数图像
Graphing functions is a core skill in IB Mathematics, requiring a deep understanding of function behaviour, transformations, and calculus. Effective graphing combines algebraic analysis with clear visual representation, and in an exam a sketch is judged on its labelled features as much as on its shape.
函数作图是IB数学的核心技能，需要深刻理解函数行为、变换和微积分。高效的作图将代数分析与清晰的视觉呈现相结合；在考试中，草图不仅看形状，更看关键特征是否标注完整。
1. Why Graphing Functions Matters | 函数作图为何重要
A graph converts algebra into geometry. Instead of manipulating symbols alone, you can see where a function is positive, where it grows fastest, how many solutions an equation has, and what happens at the edges of the domain. In IB questions, graphs are the standard tool for justifying the number of roots, for linking calculus results to shape, and for interpreting real-life models in functions, modelling and calculus topics.
图像把代数转化为几何。你不再只是操作符号，而是能直接看到函数在哪里为正、在哪里增长最快、方程有多少个解，以及定义域边缘的走势。在IB题目中，图像是论证根的个数、把微积分结论与形状对应起来、以及解释现实模型（函数、建模、微积分板块）的标准工具。
Four recurring purposes appear across the syllabus.
在整个大纲中，图像有四个反复出现的用途。
- Visualising behaviour: intercepts, turning points, asymptotes and end behaviour become visible at a glance.
可视化函数行为：截距、转折点、渐近线与末端趋势一眼可见。 - ons of f(x) = g(x), and the relative positions of the curves give the sign of f(x) - g(x).
解方程与不等式：曲线交点给出 f(x) = g(x) 的解，曲线的上下关系给出 f(x) - g(x) 的符号。 - Predicting and checking calculus: the derivative describes gradient, so the graph of f' explains the shape of f.
预判并检验微积分结果：导数描述斜率，因此 f' 的图像解释了 f 的形状。 - Communicating understanding: a clear, labelled sketch earns method and accuracy marks in papers that allow and require diagrams.
表达理解：清晰、标注完整的草图能在需要作图的题目中拿到方法与准确度分。
2. Basic Function Graphs: Linear and Quadratic | 基本函数图像：一次与二次
on is of the form x = k the graph is vertical and, strictly speaking, not a function at all because a single input would give infinitely many outputs.
线性函数表示为 f(x) = mx + c，图像是斜率为 m、y 轴截距为 c 的直线。若 m = 0 则为水平线；若方程为 x = k 的形式，图像是竖直线，严格来说并不是函数，因为一个自变量会对应无穷多个函数值。
try and the vertex lie at x = -b/(2a), the vertex y-value is c - b²/(4a), and the discriminant determines the number of x-intercepts.
二次函数形如 f(x) = ax² + bx + c (a ≠ 0)，图像为抛物线：a > 0 时开口向上，a < 0 时开口向下。对称轴与顶点横坐标由 x = -b/(2a) 给出，顶点纵坐标为 c - b²/(4a)，而判别式决定 x 轴截距的个数。
Δ = b² - 4ac
axis at one repeated root, and when Δ < 0 there are no real roots and the curve stays entirely above or below the axis.
当 Δ > 0 时抛物线与 x 轴相交两次；当 Δ = 0 时与 x 轴相切，只有一个重根；当 Δ < 0 时无实根，曲线完全在 x 轴上方或下方。
| Feature / 特征 | Quadratic f(x) = ax² + bx + c / 二次函数 |
| Direction / 开口方向 | a > 0 up, a < 0 down / a > 0 向上，a < 0 向下 |
| Vertex / 顶点 | x = -b/(2a) / 顶点横坐标 x = -b/(2a) |
| y-intercept / y 轴截距 | c / 即为 c |
| Roots / 根 | Sign of Δ / 由 Δ 的符号决定 |
3. Cubic, Quartic and Higher-Degree Polynomials | 三次、四次及更高次多项式
cubic f(x) = ax³ + bx² + cx + d has at most two turning points and at most three x-intercepts.
三次函数 f(x) = x³ 呈 S 形曲线，在原点处有一个拐点，曲线由下凹转为上凹。一般三次函数 f(x) = ax³ + bx² + cx + d 最多有两个转折点、最多三个 x 轴截距。
Higher-degree polynomials have more turning points, up to degree - 1, and at most degree x-intercepts. The leading coefficient controls end behaviour: for even degree the two ends point the same way, for odd degree they point in opposite directions.
更高次多项式具有更多的转折点，最多可达次数减 1 个，而 x 轴截距最多为次数个。首项系数决定末端趋势：偶次函数两端同向，奇次函数两端反向。
Maximum turning points = n - 1, maximum roots = n
Repeated roots change the shape near the axis: a single root cuts through, a double root touches, and a triple root crosses with a flattening - exactly what a sign diagram of the factored form tells you.
重根会改变曲线与轴接触处的形状：单根穿过 x 轴，二重根相切，三重根则穿过但在该处变平——这正是因式分解后的符号图所告诉你的信息。
4. Other Essential Standard Graphs | 其他重要的基本图像
Several non-polynomial graphs appear again and again, and their shapes must be memorised together with their asymptotes and key points. Mastery of their shapes and key features is fundamental to complex graphing.
若干非多项式图像反复出现，它们的形状必须连同渐近线与关键点一起记住。掌握它们的形状和关键特征是复杂作图的基石。
- 0 and horizontal asymptote y = 0, two separate branches in the first and third quadrants.
倒数函数：f(x) = 1/x，有竖直渐近线 x = 0 与水平渐近线 y = 0，两条分支分别位于第一、第三象限。 - Absolute value: f(x) = |x|, giving a V-shape with vertex at the origin, and y = |x - a| has its vertex at (a, 0).
绝对值函数：f(x) = |x|，呈 V 形，顶点在原点；y = |x - a| 的顶点在 (a, 0)。 - Exponential: f(x) = aˣ ing through (0, 1), with asymptote y = 0; increasing if a > 1 and decreasing if 0 < a < 1.
指数函数：f(x) = aˣ，恒过 (0, 1)，渐近线 y = 0；a > 1 时递增，0 < a < 1 时递减。 - Logarithmic: f(x) = logₐ x, the inverse of aˣ, defined for x > 0, passing through (1, 0), with asymptote x = 0.
对数函数：f(x) = logₐ x，是指数函数的反函数，定义域 x > 0，过 (1, 0)，渐近线 x = 0。 - iod 2π (360°), while tan x has period π (180°) and vertical asymptotes at x = 90° + 180°k.
三角函数：sin x 与 cos x 在 -1 与 1 之间振荡，周期 2π（360°）；tan x 周期为 π（180°），在 x = 90° + 180°k 处有竖直渐近线。
5. Domain, Range, Intercepts and Asymptotes | 定义域、值域、截距与渐近线
Before drawing anything, extract the algebraic information. Restricted domains arise from division by zero, even roots of negative numbers, and logarithms of non-positive numbers, and each restriction limits where the curve may exist. The range is then the set of y-values the curve actually reaches.
动笔之前先提取代数信息。定义域的限制来自分母为零、偶次根号内为负、以及对数真数非正；每一处限制都会限定曲线可以存在的范围，而值域则是曲线实际取到的 y 值集合。
sfy f(-x) = f(x) and are symmetric about the y-axis, while odd functions satisfy f(-x) = -f(x) and have rotational symmetry of order 2 about the origin.
y 轴截距由 x = 0 求得，x 轴截距由解方程 f(x) = 0 求得。对称性同样值得检验：偶函数满足 f(-x) = f(x)，关于 y 轴对称；奇函数满足 f(-x) = -f(x)，关于原点呈二阶旋转对称。
| Function / 函数 | Asymptotes / 渐近线 |
| 1/x | x = 0, y = 0 |
| aˣ | y = 0 |
| logₐ x | x = 0 |
| tan x | x = 90° + 180°k |
| (ax + b)/(cx + d) | x = -d/c, y = a/c |
6. Transformations of Functions | 函数变换
hifts left by h - the sign inside the bracket always works in the opposite sense to the direction of movement.
垂直平移：y = f(x) + k，当 k > 0 时图像向上平移 k，当 k < 0 时向下平移。水平平移：y = f(x - h) 向右平移 h，y = f(x + h) 向左平移 h——括号内的符号方向总与实际移动方向相反。
eflects it in the y-axis. These transformations preserve the shape but change orientation.
反射：y = -f(x) 将图像关于 x 轴反射，y = f(-x) 关于 y 轴反射。这些变换保持形状但改变方向。
Vertical stretches and shrinks: y = a f(x) with |a| > 1 stretches vertically, and 0 < |a| < 1 compresses vertically. Horizontal stretches and shrinks: y = f(bx), with |b| > 1 compresses horizontally, and 0 < |b| < 1 stretches horizontally.
垂直伸缩：y = a f(x)，当 |a| > 1 时垂直拉伸，0 < |a| < 1 时垂直压缩。水平伸缩：y = f(bx)，当 |b| > 1 时水平压缩，0 < |b| < 1 时水平拉伸。
A single transformation of each type is easy; the exam difficulty lies in reading a transformation from a completed graph, or in describing a given transformation in words with the correct factor.
单一类型的变换都不难；考试难点在于从成品图像反读变换，或用正确的伸缩系数用文字描述给定变换。
7. Combining Transformations: Order Matters | 组合变换：顺序至关重要
(b(x - h)) + k. A reliable order is: start with the stretch and reflection (which affect slopes), shift horizontally by h, then vertically scale by a, and finally shift vertically by k.
当多种变换同时作用时，使用一般形式 y = a f(b(x - h)) + k。可靠的顺序是：先做伸缩与反射（影响斜率），再水平平移 h，然后垂直按 a 缩放，最后垂直平移 k。
y = a f(b(x - h)) + k
The cleanest check is a point mapping: every point (x, y) on the original curve moves to (x/b + h, ay + k). Apply this to the vertex, the intercepts and any other known key point, and the new sketch builds itself.
最可靠的检验方式是点映射：原曲线上每一点 (x, y) 移动到 (x/b + h, ay + k)。把顶点、截距以及其他已知关键点代入，新的草图自然成形。
- Horizontal shift inside the bracket is divided by b, so a horizontal shrink by 2 also halves the visual shift.
括号内的水平平移量要除以 b，因此水平压缩 2 倍时，视觉上的平移距离也减半。 - Vertical scaling after a horizontal shift is valid because vertical operations commute with horizontal shifts.
水平平移之后再做垂直缩放是可行的，因为垂直操作与水平平移可交换。 - When describing transformations in words, state the order explicitly to earn communication marks.
用文字描述变换时，要明确写出顺序以拿沟通分。
8. Inverse Functions and Their Graphs | 反函数及其图像
Because of that reflection, the domain of f becomes the range of f⁻¹ and vice versa, and every point (x, y) on f corresponds to the point (y, x) on f⁻¹.
反函数把对应关系倒过来，因此其图像是原图像关于直线 y = x 的反射。由于这一反射，f 的定义域成为 f⁻¹ 的值域，反之亦然；f 上的点 (x, y) 对应 f⁻¹ 上的点 (y, x)。
Not every function has an inverse: f⁻¹ exists only when f is one-to-one, which is exactly the horizontal line test on the graph. If a horizontal line meets the curve more than once, the domain must be restricted before the inverse can be formed, and the restriction should be stated.
并非每个函数都有反函数：只有当 f 是一一对应时 f⁻¹ 才存在，而这正是图像上的水平线检验。若某条水平线与曲线相交多于一次，就必须先限制定义域才能求反函数，并且要写明该限制。
the functions are increasing), and applying the property f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
IB 常考任务包括：在同一坐标系中画出 f 与 f⁻¹；求它们的交点（对递增函数，交点必在 y = x 上）；以及运用 f(f⁻¹(x)) = x 与 f⁻¹(f(x)) = x 的性质。
9. Rational Functions and Curve Sketching | 有理函数与曲线作图
anishes, x = -d/c, and the horizontal asymptote is y = a/c, the ratio of leading coefficients. A sign diagram of the numerator and denominator shows which branch lies in which region.
有理函数作图前必须先分析渐近线。对 f(x) = (ax + b)/(cx + d)，竖直渐近线出现在分母为零处，即 x = -d/c；水平渐近线为 y = a/c，即首项系数之比。分子与分母的符号图则显示各分支位于哪些区域。
When the numerator has a higher degree than the denominator, perform polynomial division first. If the degree difference is exactly one, the quotient is a linear function giving an oblique (slant) asymptote that the curve approaches at infinity; if the difference is larger, the curve behaves like a higher-degree polynomial far from the origin.
当分子次数高于分母时，先做多项式除法。若次数差恰为 1，商是一次函数，给出曲线在无穷远处趋近的斜渐近线；若次数差更大，曲线在远离原点处表现为更高次多项式的形态。
A complete rational sketch should show: axes with scales, all intercepts with coordinates, all asymptotes as dashed lines with equations, the behaviour of the curve on both sides of each vertical asymptote, and the approach to the horizontal or oblique asymptote.
完整的有理函数草图应包含：带刻度的坐标轴、所有截距的坐标、所有渐近线的虚线及方程、曲线在每条竖直渐近线两侧的走向，以及向水平或斜渐近线的趋近方式。
10. Using Calculus to Sketch Curves | 用微积分作图
Calculus turns sketching from guesswork into deduction. Stationary points satisfy f'(x) = 0, and their nature is decided by the second derivative: f''(x) > 0 gives a local minimum, f''(x) < 0 gives a local maximum, and f''(x) = 0 requires a sign check of f' on both sides.
微积分把作图从猜测变成推理。驻点由 f'(x) = 0 求得，其性质由二阶导数判定：f''(x) > 0 为极小值点，f''(x) < 0 为极大值点，而 f''(x) = 0 时需要检验两侧 f' 的符号。
f'(x) = 0 → stationary point; f''(x) changes sign → point of inflection
Concavity is read from the sign of f'': positive means concave up, negative means concave down, and a change of sign indicates a point of inflection. Gradient information also gives the equations of tangents and normals at a chosen point, which are frequently requested alongside the sketch.
凹凸性由 f'' 的符号读出：为正表示下凹（上凹向上），为负表示上凸，符号改变即为拐点。导数信息还能给出指定点处的切线与法线方程，这类问题常与作图同时出现。
The reverse direction is equally examinable: given the graph of f', deduce the intervals where f increases or decreases, the locations of local extrema of f, and the concavity of f by looking at whether f' is rising or falling.
