Miscellaneous Transformations in IB Mathematics u2014 IBu6570u5b66u4e2du7684u7efcu5408u53d8u6362

Introduction to Graph Transformations — 图像变换入门

Graph transformations are one of the most versatile and frequently tested topics in IB Mathematics. They allow us to understand how changing the equation of a function affects its visual representation on the coordinate plane. The term “miscellaneous transformations” refers to the full toolkit of transformations applied to functions: translations, reflections, stretches, and combinations of these operations. Mastering these techniques is essential not only for exam success but also for developing a deeper understanding of functional relationships in calculus, modelling, and beyond.

图像变换是 IB 数学中最灵活且最常考的主题之一。它们帮助我们理解函数方程的变化如何影响其在坐标平面上的视觉表现。”综合变换”一词指的是应用于函数的完整变换工具箱：平移、反射、拉伸以及这些操作的组合。掌握这些技巧不仅对考试成功至关重要，也对在微积分、建模等领域深入理解函数关系有重要意义。

In the IB Analysis and Approaches (AA) and Applications and Interpretation (AI) courses, transformations appear across multiple syllabus topics: from basic function notation in Topic 2 (Functions) to trigonometric modelling in Topic 3 (Geometry and Trigonometry), and even in calculus when shifting graphs for integration and differentiation. Understanding how to decompose complex transformations into simpler steps is a skill that pays dividends throughout the entire course.

在 IB 分析与方法 (AA) 和应用与解释 (AI) 课程中，变换出现在多个大纲主题中：从主题 2（函数）中的基本函数符号，到主题 3（几何与三角学）中的三角函数建模，甚至在微积分中平移图像进行积分和微分时也会用到。将复杂变换分解为简单步骤的能力是一项在整个课程中都受益的技能。

Understanding Function Notation for Transformations — 理解变换的函数符号

Before diving into specific transformation types, it is crucial to understand the standard notation. When we transform a function f(x), we express the transformed function in terms of f. The general transformation form is y = a f(b(x + c)) + d, where each parameter controls a different type of transformation. This compact notation allows us to describe multiple transformations simultaneously, but it requires careful attention to the order of operations and the direction of each shift.

在深入具体变换类型之前，理解标准符号至关重要。当我们变换函数 f(x) 时，我们用 f 来表示变换后的函数。一般变换形式为 y = a f(b(x + c)) + d，其中每个参数控制一种不同类型的变换。这种紧凑的符号使我们能够同时描述多种变换，但需要仔细注意运算顺序和每次移动的方向。

Key parameters in the general form: The parameter ‘a’ represents a vertical stretch by factor |a|, with a negative value also indicating a reflection in the x-axis. The parameter ‘b’ represents a horizontal stretch by factor 1/|b|, with a negative value indicating a reflection in the y-axis. The parameter ‘c’ produces a horizontal translation of -c units (note the sign reversal, which is a common source of errors). The parameter ‘d’ produces a vertical translation of d units. In IB exams, students must be able to identify these parameters from a given equation and describe the resulting transformations in the correct sequence.

一般形式中的关键参数：参数 ‘a’ 表示以 |a| 为因子的垂直拉伸，负值还表示关于 x 轴的反射。参数 ‘b’ 表示以 1/|b| 为因子的水平拉伸，负值表示关于 y 轴的反射。参数 ‘c’ 产生 -c 单位的水平平移（注意符号反转，这是常见的错误来源）。参数 ‘d’ 产生 d 单位的垂直平移。在 IB 考试中，学生必须能够从给定方程中识别这些参数，并按正确顺序描述所产生的变换。

Vertical Translations — 垂直平移

A vertical translation shifts the entire graph up or down without changing its shape. If we add a constant k to a function, producing y = f(x) + k, every point (x, y) on the original graph moves to (x, y + k). When k is positive, the graph shifts upward; when k is negative, it shifts downward. This is perhaps the most intuitive transformation because it directly mirrors our everyday understanding of moving something “up” or “down.”

垂直平移将整个图像上下移动而不改变其形状。如果我们在函数上加一个常数 k，得到 y = f(x) + k，则原始图像上的每个点 (x, y) 移动到 (x, y + k)。当 k 为正时，图像向上移动；当 k 为负时，图像向下移动。这可能是最直观的变换，因为它直接反映了我们对”向上”或”向下”移动的日常理解。

Consider the function f(x) = x squared. The graph of y = x squared + 3 is exactly the same parabola, but its vertex has moved from (0, 0) to (0, 3). Similarly, y = x squared – 2 shifts the vertex to (0, -2). In IB problems, vertical translations often appear in the context of modelling: for example, adding a constant to a sine function shifts the midline of a wave, which can represent changing the baseline of a periodic measurement.

考虑函数 f(x) = x 平方。y = x 平方 + 3 的图像是完全相同的抛物线，但其顶点从 (0, 0) 移动到了 (0, 3)。类似地，y = x 平方 – 2 将顶点移动到 (0, -2)。在 IB 问题中，垂直平移经常出现在建模情境中：例如，在正弦函数上加一个常数会移动波的中间线，这可以表示改变周期性测量的基线。

Horizontal Translations — 水平平移

Horizontal translations shift the graph left or right. The transformation y = f(x – h) moves the graph h units to the RIGHT when h is positive, and h units to the LEFT when h is negative. This direction reversal (subtracting h moves right, adding h moves left) is consistently one of the most common mistakes students make on IB exams. A useful mnemonic is to think about what value of x makes the inside of the function equal to its original input: for y = f(x – 3), we need x = 3 to get f(0), so the point originally at x = 0 now appears at x = 3, confirming a rightward shift.

水平平移将图像向左或向右移动。变换 y = f(x – h) 当 h 为正时，将图像向右移动 h 个单位；当 h 为负时，将图像向左移动 |h| 个单位。这种方向反转（减 h 向右移，加 h 向左移）一直是学生在 IB 考试中最常见的错误之一。一个有用的记忆方法是思考什么样的 x 值能使函数内部等于其原始输入：对于 y = f(x – 3)，我们需要 x = 3 才能得到 f(0)，因此原本在 x = 0 的点现在出现在 x = 3，确认了向右的移动。

For example, starting with f(x) = x squared, the function f(x – 2) = (x – 2) squared produces a parabola whose vertex is at (2, 0). The shape is unchanged but the entire graph has been displaced two units to the right. Horizontal translations are particularly important when working with trigonometric functions, where a phase shift can dramatically alter the alignment of waves. In IB HL papers, students are often asked to determine the phase shift that makes two trigonometric graphs coincide.

例如，从 f(x) = x 平方开始，函数 f(x – 2) = (x – 2) 平方产生一个顶点在 (2, 0) 的抛物线。形状不变，但整个图像向右移动了两个单位。水平平移在处理三角函数时尤其重要，相位移动可以显著改变波的对齐方式。在 IB HL 试卷中，学生经常被要求确定使两个三角函数图像重合的相位移动。

Reflections in the Axes — 关于坐标轴的反射

Reflections flip the graph across a line, producing a mirror image. There are two fundamental reflections that every IB student must know. First, y = -f(x) reflects the graph in the x-axis: every y-coordinate is multiplied by -1, so points above the axis move below it and vice versa. Second, y = f(-x) reflects the graph in the y-axis: every x-coordinate is negated, creating a left-right mirror image. These two reflections can be combined to produce a 180-degree rotation about the origin, given by y = -f(-x).

反射将图像沿一条线翻转，产生镜像。每个 IB 学生必须掌握两种基本反射。第一，y = -f(x) 将图像关于 x 轴反射：每个 y 坐标乘以 -1，因此轴上方的点移到下方，反之亦然。第二，y = f(-x) 将图像关于 y 轴反射：每个 x 坐标取反，创建左右镜像。这两种反射可以组合产生关于原点的 180 度旋转，由 y = -f(-x) 表示。

Even and odd functions provide a natural testing ground for understanding reflections. An even function satisfies f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Examples include f(x) = x squared, f(x) = cos x, and f(x) = |x|. An odd function satisfies f(-x) = -f(x), meaning its graph has 180-degree rotational symmetry about the origin. Examples include f(x) = x cubed, f(x) = sin x, and f(x) = 1/x. IB questions frequently ask students to determine whether a given function is even, odd, or neither by testing the algebraic definitions.

偶函数和奇函数为理解反射提供了天然的测试场。偶函数在其定义域内满足 f(-x) = f(x)，意味着其图像关于 y 轴对称。例子包括 f(x) = x 平方、f(x) = cos x 和 f(x) = |x|。奇函数满足 f(-x) = -f(x)，意味着其图像关于原点有 180 度旋转对称。例子包括 f(x) = x 立方、f(x) = sin x 和 f(x) = 1/x。IB 题目经常要求学生通过测试代数定义来判断给定函数是偶函数、奇函数还是两者都不是。

Vertical and Horizontal Stretches — 垂直和水平拉伸

Stretches change the shape of a graph by scaling it in one direction. A vertical stretch multiplies all y-coordinates by a factor: y = a f(x) stretches the graph vertically by factor a. When |a| is greater than 1, the graph becomes taller; when |a| is less than 1 (but not zero), the graph is compressed vertically. A negative value of a also includes a reflection in the x-axis, making y = a f(x) a combined transformation of vertical stretch and possible reflection.

拉伸通过在一个方向上缩放来改变图像的形状。垂直拉伸将所有 y 坐标乘以一个因子：y = a f(x) 以因子 a 垂直拉伸图像。当 |a| 大于 1 时，图像变高；当 |a| 小于 1（但不为零）时，图像被垂直压缩。a 的负值还包括关于 x 轴的反射，使得 y = a f(x) 成为垂直拉伸和可能反射的组合变换。

Horizontal stretches are controlled by the parameter b in y = f(bx). The graph stretches horizontally by factor 1/|b|. When |b| is greater than 1, the graph is compressed horizontally (squeezed inward); when |b| is less than 1, the graph is stretched horizontally (pulled outward). The reciprocal relationship between the parameter and the visual effect is another common point of confusion. Students should practise identifying stretches from equations such as y = f(2x) (horizontal compression by factor 1/2) or y = f(x/3) (horizontal stretch by factor 3).

水平拉伸由 y = f(bx) 中的参数 b 控制。图像以因子 1/|b| 水平拉伸。当 |b| 大于 1 时，图像被水平压缩（向内挤压）；当 |b| 小于 1 时，图像被水平拉伸（向外拉开）。参数与视觉效果之间的倒数关系是另一个常见的混淆点。学生应练习从方程中识别拉伸，如 y = f(2x)（以因子 1/2 水平压缩）或 y = f(x/3)（以因子 3 水平拉伸）。

Combining Multiple Transformations — 组合多种变换

In most IB exam questions, transformations do not appear in isolation. Students must handle functions of the form y = a f(b(x + c)) + d, which combines up to four separate transformations. The key to success lies in the correct order of operations. When applying transformations to sketch a graph from a parent function, the recommended sequence is: first, horizontal translation (parameter c); second, horizontal stretch (parameter b); third, reflection and vertical stretch (parameter a); and fourth, vertical translation (parameter d). This order follows the principle of working from the inside of the function outward.

在大多数 IB 考题中，变换不会单独出现。学生必须处理形如 y = a f(b(x + c)) + d 的函数，它组合了最多四种不同的变换。成功的关键在于正确的操作顺序。当应用变换从父函数绘制图像时，推荐顺序是：第一，水平平移（参数 c）；第二，水平拉伸（参数 b）；第三，反射和垂直拉伸（参数 a）；第四，垂直平移（参数 d）。这个顺序遵循从函数内部向外工作的原则。

A common IB question format presents a transformed function and asks students to describe the sequence of transformations that maps f(x) onto the new function. For example, to transform f(x) = x squared into g(x) = -2(x + 3) squared + 5, the correct description is: translate 3 units left, stretch vertically by factor 2, reflect in the x-axis, and translate 5 units up. Notice how the horizontal translation (inside the bracket) happens first, and the vertical translation (outside the bracket) happens last. Reversing this order typically produces a different result, which is why precision matters.

一个常见的 IB 题目格式是给出变换后的函数，要求学生描述将 f(x) 映射到新函数的变换序列。例如，要将 f(x) = x 平方 变换为 g(x) = -2(x + 3) 平方 + 5，正确的描述是：向左平移 3 个单位，垂直拉伸因子 2，关于 x 轴反射，向上平移 5 个单位。注意水平平移（括号内）最先发生，垂直平移（括号外）最后发生。颠倒这个顺序通常会产生不同的结果，这就是精确性很重要的原因。

Transformations of Specific Function Families — 特定函数族的变换

While the principles of transformation apply universally, each function family exhibits unique characteristics under transformation that are worth studying individually. For quadratic functions f(x) = ax squared + bx + c, completing the square reveals the vertex form f(x) = a(x – h) squared + k, which directly encodes a horizontal translation by h, a vertical translation by k, and a vertical stretch by a. This connection between algebraic manipulation and geometric transformation is a recurring theme in IB Mathematics.

虽然变换原理普遍适用，但每个函数族在变换下表现出独特的特征，值得单独研究。对于二次函数 f(x) = ax 平方 + bx + c，配方可以揭示顶点形式 f(x) = a(x – h) 平方 + k，它直接编码了水平平移 h、垂直平移 k 和垂直拉伸 a。这种代数操作与几何变换之间的联系是 IB 数学中反复出现的主题。

Exponential functions of the form f(x) = a times b to the power of (x – h) plus k demonstrate how transformations affect asymptotic behaviour. The horizontal asymptote, originally y = 0 for the basic exponential function, shifts to y = k after a vertical translation. The horizontal translation parameter h shifts the curve left or right without changing the asymptote’s vertical position. IB students should be able to sketch exponential graphs showing the correct asymptote, intercepts, and general shape after multiple transformations.

形如 f(x) = a 乘以 b 的 (x – h) 次方 加 k 的指数函数展示了变换如何影响渐近行为。水平渐近线，原本对于基本指数函数是 y = 0，在垂直平移后移动到 y = k。水平平移参数 h 将曲线向左或向右移动而不改变渐近线的垂直位置。IB 学生应该能够在多重变换后绘制指数函数图像，显示正确的渐近线、截距和大致形状。

Trigonometric functions present the richest ground for transformation practice. The general sine function y = A sin(B(x + C)) + D introduces amplitude (A), period (2 pi divided by B), phase shift (C), and vertical shift (D). The same structure applies to cosine and tangent functions with appropriate adjustments. IB exam questions frequently ask students to determine the equation of a trigonometric graph from its visual features, or to sketch a transformed trigonometric function showing key points such as maxima, minima, and intercepts within a specified domain.

三角函数为变换练习提供了最丰富的土壤。一般正弦函数 y = A sin(B(x + C)) + D 引入了振幅 (A)、周期 (2 pi 除以 B)、相位移动 (C) 和垂直移动 (D)。同样的结构适用于余弦和正切函数，只需适当调整。IB 考题经常要求学生从视觉特征确定三角函数图像的方程，或在指定定义域内绘制变换后的三角函数图像，显示最大值、最小值和截距等关键点。

Inverse Functions and Transformation Relationships — 反函数与变换关系

The relationship between a function and its inverse provides another perspective on transformations. The graph of an inverse function is the reflection of the original function in the line y = x. This means that every point (a, b) on the graph of f(x) corresponds to the point (b, a) on the graph of f inverse (x). This transformation property can be used to quickly sketch inverse functions once the original function is known, and it explains why the domain and range swap between a function and its inverse.

函数与其反函数之间的关系为变换提供了另一个视角。反函数的图像是原函数关于直线 y = x 的反射。这意味着 f(x) 图像上的每个点 (a, b) 对应 f 反函数 (x) 图像上的点 (b, a)。这个变换性质可以用来在已知原函数的情况下快速绘制反函数图像，也解释了为什么定义域和值域在函数与其反函数之间互换。

Not all functions have inverses that are themselves functions. For a function to have an inverse function, it must be one-to-one (injective): each y-value in the range must correspond to exactly one x-value in the domain. The horizontal line test provides a visual way to check this: if any horizontal line intersects the graph more than once, the function is not one-to-one and must have its domain restricted to create an inverse function. IB questions often ask students to find the largest possible domain for which a given function has an inverse, testing understanding of both function properties and transformation geometry.

并非所有函数都有本身是函数的反函数。一个函数要有反函数，它必须是一一对应的（单射）：值域中的每个 y 值必须对应定义域中恰好一个 x 值。水平线测试提供了检查这一点的可视化方法：如果任何水平线与图像相交超过一次，则该函数不是一一对应的，必须限制其定义域才能创建反函数。IB 题目经常要求学生找出给定函数存在反函数的最大可能定义域，测试对函数性质和变换几何的理解。

Analysing Transformations from Given Graphs — 从给定图像分析变换

A significant portion of IB transformation questions requires students to work backwards: given the graphs of a parent function and a transformed function, determine the transformation parameters. This reverse-engineering approach tests deeper understanding because students must deduce the sequence of operations from visual evidence. Key strategies include identifying the positions of critical points (turning points, intercepts, asymptotes) in both graphs and computing the differences.

IB 变换题目的一个重要部分要求学生反向工作：给定父函数和变换后函数的图像，确定变换参数。这种逆向工程方法测试更深层次的理解，因为学生必须从视觉证据中推断操作序列。关键策略包括识别两个图像中关键点（转折点、截距、渐近线）的位置并计算差异。

For example, if a parabola with vertex at (-1, 4) is shown to be a transformation of f(x) = x squared, the student can determine that a horizontal translation of -1 (or +1 inside the function) and a vertical translation of +4 have been applied. If the parabola also opens downward and is narrower than the parent function, a negative vertical stretch with |a| greater than 1 is also present. This systematic approach of isolating one transformation at a time, starting with the most visually obvious changes, is the most reliable method.

例如，如果顶点在 (-1, 4) 的抛物线被显示为 f(x) = x 平方 的变换，学生可以确定应用了 -1 的水平平移（或函数内部的 +1）和 +4 的垂直平移。如果抛物线还向下开口且比父函数更窄，则还存在 |a| 大于 1 的负垂直拉伸。这种从最明显的视觉变化开始，一次隔离一个变换的系统方法是 最可靠的方法。

Common Mistakes and How to Avoid Them — 常见错误及如何避免

Several misconceptions about transformations persist even among strong IB students. The most frequent error involves the direction of horizontal translations: mistaking f(x + 2) for a rightward shift when it actually moves the graph left by 2. A reliable check is to evaluate the function at a specific point: if the original function passes through (0, 0), then f(x + 2) passes through (-2, 0), confirming a leftward shift. Another common error is confusing the order of horizontal stretch and horizontal translation inside the bracket.

即使在优秀的 IB 学生中，也有几种关于变换的误解持续存在。最常见的错误涉及水平平移的方向：将 f(x + 2) 误认为是向右移动，而实际上它将图像向左移动了 2。一个可靠的检查方法是在特定点评估函数：如果原函数经过 (0, 0)，那么 f(x + 2) 经过 (-2, 0)，确认了向左的移动。另一个常见错误是混淆括号内水平拉伸和水平平移的顺序。

Students also frequently misidentify the period of transformed trigonometric functions. For y = sin(2x), the period is pi, not 2 pi (the period of sin x). The formula period = 2 pi divided by |b| must be applied correctly. Another pitfall involves the amplitude of functions like y = 3 sin x + 1, where students sometimes incorrectly state the amplitude as 4 instead of 3, confusing the vertical shift with the amplitude. The amplitude is simply the coefficient of the trigonometric term, independent of any vertical shift.

学生还经常错误识别变换后三角函数的周期。对于 y = sin(2x)，周期是 pi，而不是 2 pi（sin x 的周期）。公式 周期 = 2 pi 除以 |b| 必须正确应用。另一个陷阱涉及像 y = 3 sin x + 1 这样的函数的振幅，学生有时错误地将振幅说成 4 而不是 3，混淆了垂直移动与振幅。振幅仅仅是三角函数项的系数，与任何垂直移动无关。

Transformations in the Context of Calculus — 微积分背景下的变换

Transformations play a significant role in IB calculus problems, particularly in understanding how derivatives and integrals behave under transformation. If g(x) = f(x) + k, then g'(x) = f'(x): vertical translations do not affect the derivative, which makes intuitive sense because the slope of a curve does not change when the entire curve is shifted up or down. However, horizontal translations do change the derivative because the slope is being evaluated at a different x-value: if g(x) = f(x – h), then g'(x) = f'(x – h).

变换在 IB 微积分问题中扮演着重要角色，特别是在理解导数和积分在变换下的行为方面。如果 g(x) = f(x) + k，那么 g'(x) = f'(x)：垂直平移不影响导数，这在直觉上是有道理的，因为当整个曲线上下移动时，曲线的斜率不会改变。然而，水平平移确实会改变导数，因为斜率是在不同的 x 值处评估的：如果 g(x) = f(x – h)，那么 g'(x) = f'(x – h)。

For stretches, the relationship is multiplicative. If g(x) = a f(x), then g'(x) = a f'(x): vertical stretches multiply the derivative by the same factor. If g(x) = f(bx), then g'(x) = b f'(bx) by the chain rule: horizontal compressions amplify the derivative. These connections between transformations and rates of change are particularly relevant in optimisation problems and kinematic modelling, where understanding how shifting or scaling a position function affects velocity and acceleration is essential.

对于拉伸，关系是乘性的。如果 g(x) = a f(x)，那么 g'(x) = a f'(x)：垂直拉伸以相同的因子乘以导数。如果 g(x) = f(bx)，那么根据链式法则 g'(x) = b f'(bx)：水平压缩会放大导数。这些变换与变化率之间的联系在优化问题和运动学建模中尤为相关，在这些问题中，理解移动或缩放位置函数如何影响速度和加速度是必不可少的。

Exam Strategies and Practice Techniques — 考试策略与练习技巧

Success with transformation questions on IB exams requires both conceptual understanding and efficient problem-solving techniques. When faced with a multi-step transformation problem, start by identifying the parent function and writing it explicitly. Then isolate each transformation parameter one at a time, working from the inside of the function outward. For sketching problems, plot the key points of the parent function first, then apply each transformation to those points sequentially rather than trying to imagine the entire transformed graph at once.

在 IB 考试中成功应对变换题目需要概念理解和高效的解题技巧。面对多步变换问题时，从识别父函数并明确写出开始。然后一次隔离一个变换参数，从函数内部向外工作。对于绘图问题，先标出父函数的关键点，然后依次对每个点应用变换，而不是试图一次性想象整个变换后的图像。

Time management is critical in IB exams. Transformation questions often appear in Section A (non-calculator) where algebraic precision is paramount. Practise manipulating the transformation form y = a f(b(x + c)) + d until you can identify parameters almost automatically. For Section B (calculator) questions involving real-world data, focus on recognising which transformation best fits the modelling scenario: does the data suggest a vertical shift to adjust a baseline, a horizontal shift to adjust timing, or a stretch to adjust the rate of growth or oscillation?

时间管理在 IB 考试中至关重要。变换题目经常出现在 A 部分（不允许使用计算器），其中代数精确性至关重要。练习操作变换形式 y = a f(b(x + c)) + d，直到你能几乎自动地识别参数。对于涉及真实世界数据的 B 部分（允许使用计算器）题目，专注于识别哪种变换最适合建模场景：数据是否建议使用垂直移动来调整基线，水平移动来调整时间，还是拉伸来调整增长或振荡的速率？

Summary and Key Takeaways — 总结与关键要点

Transformations form a unifying thread throughout the IB Mathematics curriculum. The four fundamental types are translation (vertical and horizontal), reflection (in both axes), stretch (vertical and horizontal), and combinations thereof, all encapsulated in the general form y = a f(b(x + c)) + d. The most important practical skill is the ability to apply these transformations in the correct order: horizontal translation, horizontal stretch, reflection and vertical stretch, and finally vertical translation.

变换构成了贯穿 IB 数学课程的统一点。四种基本类型是平移（垂直和水平）、反射（关于两个轴）、拉伸（垂直和水平）及其组合，所有这些都包含在一般形式 y = a f(b(x + c)) + d 中。最重要的实践技能是能够按正确顺序应用这些变换：水平平移、水平拉伸、反射和垂直拉伸，最后是垂直平移。

For IB students aiming for high marks, mastery of transformations provides a competitive edge across multiple assessment objectives. Whether analysing the properties of polynomial, exponential, or trigonometric functions, solving optimisation problems in calculus, or constructing mathematical models from real data, the ability to visualise and manipulate transformations is indispensable. Regular practice with past paper questions, particularly those that combine transformations with other topics like calculus or trigonometry, will build the fluency needed for exam success.

对于追求高分的 IB 学生来说，掌握变换在多个评估目标中提供了竞争优势。无论是分析多项式、指数或三角函数的性质，解决微积分中的优化问题，还是从真实数据构建数学模型，可视化和操作变换的能力都是不可或缺的。定期练习历年真题，特别是那些将变换与微积分或三角学等其他主题结合的题目，将培养学生考试成功所需的熟练程度。