📚 Features, Similarities and Differences of Functions | 函数的特征、相似性与差异
In Edexcel A Level Mathematics, many exam questions ask you to describe and compare the features of different functions. These features include intercepts, roots, asymptotes, symmetry, periodicity, domain and range. Being able to identify similarities and differences helps you sketch graphs and solve problems confidently.
在Edexcel A Level数学中，许多考题要求你描述并比较不同函数的特征。这些特征包括截距、根、渐近线、对称性、周期、定义域和值域。能够识别相似性和差异性有助于你自信地画图并解决问题。
1. Key Features of a Graph | 图形的关键特征
When exam questions refer to ‘features’, they usually mean the main characteristics that define a graph. These include where the graph crosses the axes, where it turns, how it behaves at extremes, and whether it repeats.
当考题提到“特征”时，通常指的是定义一个图形的主要特性。这些包括图形与坐标轴的交点、转折点、在极端处的行为，以及它是否重复。
Similarities are often found within the same family of functions, while differences appear when families are compared. For example, all quadratic graphs share a parabolic shape, but their roots and turning points may differ.
相似性通常出现在同一函数族中，而差异性则在比较不同函数族时出现。例如，所有二次函数图像都共享抛物线形状，但它们的根和转折点可能不同。
2. Intercepts and Roots | 截距与根
The y-intercept is found by substituting x = 0 into the function. For a polynomial y = f(x), the constant term often gives the y-intercept directly.
y轴截距通过将x = 0代入函数求得。对于多项式y = f(x)，常数项通常直接给出y轴截距。
Roots or x-intercepts are found by solving f(x) = 0. Quadratic functions can have 0, 1 or 2 real roots, while cubic functions can have 1, 2 or 3 real roots. Comparing these shows a clear difference in maximum possible roots.
根或x轴截距通过解方程f(x) = 0求得。二次函数可以有0、1或2个实根，而三次函数可以有1、2或3个实根。比较这些可以看出最大可能根数的明显差异。
A linear function has exactly one root unless it is horizontal. A reciprocal function of the form 1/x has no x-intercept because 1/x = 0 has no solution.
线性函数除非是水平的，否则恰好有一个根。形如1/x的反比例函数没有x轴截距，因为1/x = 0无解。
3. Asymptotes and End Behaviour | 渐近线与末端行为
Reciprocal functions such as y = 1/x have a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. These lines are approached but never crossed by the graph.
反比例函数如y = 1/x在x = 0处有垂直渐近线，在y = 0处有水平渐近线。图像无限接近这些直线但不会穿过它们。
Exponential functions y = aˣ with a > 1 have a horizontal asymptote at y = 0 as x → −∞, while logarithmic functions y = ln x have a vertical asymptote at x = 0. This is a key difference in end behaviour.
底数a > 1的指数函数y = aˣ在x → −∞时以y = 0为水平渐近线，而对数函数y = ln x在x = 0处有垂直渐近线。这是末端行为的关键差异。
Polynomial functions do not have asymptotes; their end behaviour is determined by the leading term. For even degree with positive leading coefficient, both ends rise to +∞.
多项式函数没有渐近线；它们的末端行为由首项决定。对于偶次且首项系数为正的多项式，两端都趋向+∞。
4. Symmetry: Even and Odd Functions | 对称性：偶函数与奇函数
A function is even if f(−x) = f(x) for all x, giving symmetry about the y-axis. Examples include y = x², y = cos x and y = |x|.
如果对于所有x都有f(−x) = f(x)，则函数是偶函数，图像关于y轴对称。例子包括y = x²、y = cos x和y = |x|。
A function is odd if f(−x) = −f(x), giving 180° rotational symmetry about the origin. Examples include y = x³, y = sin x and y = tan x.
如果对于所有x都有f(−x) = −f(x)，则函数是奇函数，图像关于原点具有180°旋转对称性。例子包括y = x³、y = sin x和y = tan x。
Comparing y = x² and y = x³ illustrates this clearly: the parabola is symmetric in the y-axis, while the cubic has rotational symmetry. Many functions are neither even nor odd, such as y = x² + x.
比较y = x²和y = x³可以清楚地说明这一点：抛物线关于y轴对称，而三次函数具有旋转对称性。许多函数既不是偶函数也不是奇函数，例如y = x² + x。
5. Periodicity and Repetition | 周期性与重复性
Trigonometric functions are periodic: they repeat their values at regular intervals. For sin x and cos x the period is 2π, while for tan x the period is π.
三角函数是周期性的：它们以固定间隔重复取值。对于sin x和cos x，周期为2π；对于tan x，周期为π。
Polynomial, exponential and logarithmic functions are not periodic. This difference is important when comparing trigonometric graphs with other families.
多项式、指数和对数函数不是周期性的。这一差异在比较三角函数图像与其他函数族时非常重要。
Within trigonometric graphs, sin x and cos x have the same amplitude and period, but y = cos x is a horizontal translation of y = sin x by π/2 to the left. This is a similarity with a phase difference.
在三角函数图像中，sin x和cos x具有相同的振幅和周期，但y = cos x是y = sin x向左平移π/2得到的。这是一个具有相位差的相似性。
6. Domain and Range | 定义域与值域
The domain is the set of all possible input values x for which the function is defined. The range is the set of all possible output values f(x).
定义域是函数有定义的所有可能输入值x的集合。值域是所有可能输出值f(x)的集合。
For y = x², the domain is all real numbers, but the range is y ≥ 0. For y = √x, the domain is x ≥ 0 and the range is y ≥ 0. These restrictions can cause confusion in comparisons.
对于y = x²，定义域为所有实数，但值域为y ≥ 0。对于y = √x，定义域为x ≥ 0，值域为y ≥ 0。这些限制在比较时容易混淆。
Exponential functions y = aˣ have domain all real numbers and range y > 0. Logarithmic functions y = logₐ x have domain x > 0 and range all real numbers. They are inverse functions, so their domains and ranges are swapped.
指数函数y = aˣ的定义域为所有实数，值域为y > 0。对数函数y = logₐ x的定义域为x > 0，值域为所有实数。它们互为反函数，因此定义域和值域互换。
7. Comparing Polynomial Functions | 比较多项式函数
Linear functions y = mx + c have constant gradient m. Quadratic functions y = ax² + bx + c have one turning point. Cubic functions y = ax³ + bx² + cx + d can have up to two turning points.
线性函数y = mx + c具有恒定的斜率m。二次函数y = ax² + bx + c有一个转折点。三次函数y = ax³ + bx² + cx + d最多有两个转折点。
The degree of a polynomial determines the maximum number of roots and turning points. A polynomial of degree n has at most n real roots and at most n − 1 turning points.
多项式的次数决定了根和转折点的最大数量。n次多项式最多有n个实根和最多n − 1个转折点。
All polynomials have smooth, continuous graphs with no asymptotes or gaps. This is a key similarity across polynomial functions, unlike rational functions which may have asymptotes.
所有多项式函数都有光滑、连续的图像，没有渐近线或间断点。这是多项式函数之间的关键相似性，与可能有渐近线的有理函数不同。
8. Comparing Trigonometric Functions | 比较三角函数
y = sin x and y = cos x both have period 2π, range [−1, 1], and are continuous for all real x. Their similarities make them easy to compare and transform.
y = sin x和y = cos x都有周期2π、值域[−1, 1]，并且在所有实数x上连续。它们的相似性使比较和变换变得容易。
y = tan x has period π, range all real numbers, and vertical asymptotes at x = π/2 + nπ. This differs greatly from sin and cos, which have no asymptotes.
y = tan x的周期为π，值域为所有实数，并且在x = π/2 + nπ处有垂直渐近线。这与没有渐近线的sin和cos差异很大。
The sine curve starts at 0 and rises, while the cosine curve starts at 1. A phase shift of π/2 converts one into the other: cos x = sin(x + π/2).
正弦曲线从0开始上升，而余弦曲线从1开始。π/2的相位差可以将一个转换为另一个：cos x = sin(x + π/2)。
9. Comparing Exponential and Logarithmic Functions | 比较指数函数与对数函数
Exponential functions y = aˣ with a > 1 grow very rapidly for positive x and decay towards zero for negative x. Their graphs always pass through (0, 1).
底数a > 1的指数函数y = aˣ在正x处增长非常快，在负x处衰减到零。它们的图像总是经过点(0, 1)。
Logarithmic functions y = logₐ x grow very slowly for large x and are not defined for x ≤ 0. Their graphs always pass through (1, 0).
对数函数y = logₐ x对于大x增长非常缓慢，并且在x ≤ 0时没有定义。它们的图像总是经过点(1, 0)。
The exponential and logarithmic graphs are reflections of each other in the line y = x because they are inverse functions. This is a powerful similarity used in solving equations.
指数函数和对数函数的图像关于直线y = x对称，因为它们互为反函数。这是解方程时非常有用的一种相似性。
10. Transformations and Feature Changes | 变换与特征变化
Transformations change features in predictable ways. A translation by vector (a, b) shifts every point and therefore moves intercepts and asymptotes by the same amount.
变换以可预测的方式改变特征。平移向量(a, b)会移动每一个点，因此也以相同
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