本讲义涵盖主题 2,代数和图像(Algebra and graphs)。标记 (Extended) 的部分只在拓展卷上考查;其他一切对两个层次都适用。
代数与图像
IGCSE 数学 · 第 2 主题
2.1
代数导论
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Know that letters can be used to represent generalised numbers. | |
| 2 Substitute numbers into expressions and formulas. |
来源:剑桥国际大纲
在代数(algebra)里我们用字母代表数。一个值能变化的字母是一个变量(variable)。代入(substitute)意味着把一个数放在一个字母的位置。
Worked example. 当 $x = 4$ 和 $y = 5$ 时求 $3x^{2} - 2y$ 的值。
$$3 \times 4^{2} - 2 \times 5 = 3 \times 16 - 10 = 48 - 10 = 38.$$
探索
Algebra manipulation route
Follow expression work from collecting terms to solving.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| algebra | 代数 | dài shù |
| variable | 变量 | biàn liàng |
| substitute | 代入 | dài rù |
2.2
代数式的运算
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Simplify expressions by collecting like terms. | Simplify means give the answer in its simplest form, e.g. $2a + 3b + 5a - 9b = 7a - 6b$. |
| 2 Expand products of algebraic expressions. | e.g. expand $3x(2x - 4y)$. Includes products of two brackets involving one variable, e.g. expand $(2x + 1)(x - 4)$. |
| 3 Factorise by extracting common factors. | Factorise means factorise fully, e.g. $9x^2 + 15xy = 3x(3x + 5y)$. |
| Subject content | Notes and examples |
|---|---|
| 1 Simplify expressions by collecting like terms. | Simplify means give the answer in its simplest form, e.g. $2a^2 + 3ab - 1 + 5a^2 - 9ab + 4 = 7a^2 - 6ab + 3$. |
| 2 Expand products of algebraic expressions. | e.g. expand $3x(2x - 4y)$, $(3x + y)(x - 4y)$. Includes products of more than two brackets, e.g. expand $(x - 2)(x + 3)(2x + 1)$. |
| 3 Factorise by extracting common factors. | Factorise means factorise fully, e.g. $9x^2 + 15xy = 3x(3x + 5y)$. |
| 4 Factorise expressions of the form: • $ax + bx + kay + kby$ • $a^2x^2 - b^2y^2$ • $a^2 + 2ab + b^2$ • $ax^2 + bx + c$ • $ax^3 + bx^2 + cx$. | |
| 5 Complete the square for expressions in the form $ax^2 + bx + c$. |
来源:剑桥国际大纲
一个项(term)是一个表达式(expression)的单个部分,例如 $5a$ 或 $-9b$。同类项(like terms)有恰好相同的字母;你可以加或减它们。字母前面的数是系数(coefficient)。
Worked example. 化简 $2a^{2} + 3ab - 1 + 5a^{2} - 9ab + 4$。
收集同类项:$2a^{2} + 5a^{2} = 7a^{2}$,$3ab - 9ab = -6ab$,$-1 + 4 = 3$。所以答案是
$$7a^{2} - 6ab + 3.$$
展开(expand)意味着把括号(brackets)乘出来。把里面的每一项乘以外面的项;对于两个括号,把第一个里的每一项乘以第二个里的每一项。
Worked examples.
$$3x(2x - 4y) = 6x^{2} - 12xy.$$
$$(2x + 1)(x - 4) = 2x^{2} - 8x + x - 4 = 2x^{2} - 7x - 4.$$
对于三个括号(拓展(Extended)),先展开两个,然后乘以第三个:
$$(x - 2)(x + 3)(2x + 1) = (x^{2} + x - 6)(2x + 1) = 2x^{3} + 3x^{2} - 11x - 6.$$
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| term | 项 | xiàng |
| expression | 表达式 | biǎo dá shì |
| like terms | 同类项 | tóng lèi xiàng |
| coefficient | 系数 | xì shù |
| expand | 展开 | zhǎn kāi |
| brackets | 括号 | kuò hào |
| area | 面积 | miàn jī |
2.2
代数式的运算
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Simplify expressions by collecting like terms. | Simplify means give the answer in its simplest form, e.g. $2a + 3b + 5a - 9b = 7a - 6b$. |
| 2 Expand products of algebraic expressions. | e.g. expand $3x(2x - 4y)$. Includes products of two brackets involving one variable, e.g. expand $(2x + 1)(x - 4)$. |
| 3 Factorise by extracting common factors. | Factorise means factorise fully, e.g. $9x^2 + 15xy = 3x(3x + 5y)$. |
| Subject content | Notes and examples |
|---|---|
| 1 Simplify expressions by collecting like terms. | Simplify means give the answer in its simplest form, e.g. $2a^2 + 3ab - 1 + 5a^2 - 9ab + 4 = 7a^2 - 6ab + 3$. |
| 2 Expand products of algebraic expressions. | e.g. expand $3x(2x - 4y)$, $(3x + y)(x - 4y)$. Includes products of more than two brackets, e.g. expand $(x - 2)(x + 3)(2x + 1)$. |
| 3 Factorise by extracting common factors. | Factorise means factorise fully, e.g. $9x^2 + 15xy = 3x(3x + 5y)$. |
| 4 Factorise expressions of the form: • $ax + bx + kay + kby$ • $a^2x^2 - b^2y^2$ • $a^2 + 2ab + b^2$ • $ax^2 + bx + c$ • $ax^3 + bx^2 + cx$. | |
| 5 Complete the square for expressions in the form $ax^2 + bx + c$. |
来源:剑桥国际大纲
因式分解(factorise)是展开的反面:把表达式写成括号的一个乘积。总是先取出公因式(common factor)。
Worked example. $9x^{2} + 15xy = 3x(3x + 5y)$,因为 $3x$ 整除两项。
下面的模式是拓展(Extended)。
分组(grouping)(四项):从每一对取出一个公因式。
$$xy + 2x + 3y + 6 = x(y + 2) + 3(y + 2) = (x + 3)(y + 2).$$
平方差(difference of two squares):$a^{2} - b^{2} = (a + b)(a - b)$。
$$9x^{2} - 16 = (3x + 4)(3x - 4).$$
完全平方(perfect square):$a^{2} + 2ab + b^{2} = (a + b)^{2}$。
$$x^{2} + 6x + 9 = (x + 3)^{2}.$$
二次(quadratic)表达式 $ax^{2} + bx + c$:找到两个相乘得 $a \times c$ 而相加得 $b$ 的数,然后拆分中间项。
Worked example. 因式分解 $2x^{2} + 7x + 3$。
这里 $a \times c = 6$ 而 $b = 7$。数 $1$ 和 $6$ 起作用。拆分并分组:
$$2x^{2} + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1).$$
对于 $ax^{3} + bx^{2} + cx$,先取出公共的 $x$:$2x^{3} + 7x^{2} + 3x = x(2x^{2} + 7x + 3) = x(x + 3)(2x + 1)$。
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| factorise | 因式分解 | yīn shì fēn jiě |
| common factor | 公因式 | gōng yīn shì |
| difference of two squares | 平方差 | píng fāng chà |
| perfect square | 完全平方 | wán quán píng fāng |
| quadratic | 二次 | èr cì |
2.2
代数式的运算
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Simplify expressions by collecting like terms. | Simplify means give the answer in its simplest form, e.g. $2a^2 + 3ab - 1 + 5a^2 - 9ab + 4 = 7a^2 - 6ab + 3$. |
| 2 Expand products of algebraic expressions. | e.g. expand $3x(2x - 4y)$, $(3x + y)(x - 4y)$. Includes products of more than two brackets, e.g. expand $(x - 2)(x + 3)(2x + 1)$. |
| 3 Factorise by extracting common factors. | Factorise means factorise fully, e.g. $9x^2 + 15xy = 3x(3x + 5y)$. |
| 4 Factorise expressions of the form: • $ax + bx + kay + kby$ • $a^2x^2 - b^2y^2$ • $a^2 + 2ab + b^2$ • $ax^2 + bx + c$ • $ax^3 + bx^2 + cx$. | |
| 5 Complete the square for expressions in the form $ax^2 + bx + c$. |
来源:剑桥国际大纲
配方法(completing the square)把 $x^{2} + bx + c$ 重写为 $(x + p)^{2} + q$。取 $x$ 系数的一半、把它平方,然后平衡。
Worked example. 把 $x^{2} + 6x + 1$ 写成配方形式。
$6$ 的一半是 $3$,而 $3^{2} = 9$:
$$x^{2} + 6x + 1 = (x + 3)^{2} - 9 + 1 = (x + 3)^{2} - 8.$$
当 $x^{2}$ 有一个系数时,先把它从前两项取出:
$$2x^{2} + 8x + 3 = 2(x^{2} + 4x) + 3 = 2\big((x + 2)^{2} - 4\big) + 3 = 2(x + 2)^{2} - 5.$$
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| completing the square | 配方法 | pèi fāng fǎ |
2.3
代数分式
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Manipulate algebraic fractions. | Examples include: • $\frac{x}{3} + \frac{x - 4}{2}$ • $\frac{2x}{3} - \frac{3(x - 5)}{2}$ • $\frac{3a}{4} \times \frac{9a}{10}$ • $\frac{3a}{4} \div \frac{9a}{10}$ • $\frac{1}{x - 2} + \frac{x + 1}{x - 3}$. |
| 2 Factorise and simplify rational expressions. | e.g. $\frac{x^2 - 2x}{x^2 - 5x + 6}$. |
来源:剑桥国际大纲
一个分式(algebraic fraction)在上部或底部有代数。用一个公分母加和减;像普通分数一样乘和除。
Worked examples.
$$\frac{x}{3} + \frac{x - 4}{2} = \frac{2x}{6} + \frac{3(x - 4)}{6} = \frac{2x + 3x - 12}{6} = \frac{5x - 12}{6}.$$
$$\frac{3a}{4} \div \frac{9a}{10} = \frac{3a}{4} \times \frac{10}{9a} = \frac{30a}{36a} = \frac{5}{6}.$$
要化简一个有理式(rational expression),因式分解上部和底部,然后约去公共的括号。
$$\frac{x^{2} - 2x}{x^{2} - 5x + 6} = \frac{x(x - 2)}{(x - 2)(x - 3)} = \frac{x}{x - 3}.$$
探索
Algebraic fraction route
Simplify algebraic fractions by factorising before cancelling.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| algebraic fraction | 分式 | fēn shì |
| rational expression | 有理式 | yǒu lǐ shì |
2.4
指数 II
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Understand and use indices (positive, zero and negative). | e.g. $2^x = 32$. Find the value of $x$. |
| 2 Understand and use the rules of indices. | e.g. simplify: • $(5x^3)^2$ • $12a^5 \div 3a^{-2}$ • $6x^7y^4 \times 5x^{-5}y$. Knowledge of logarithms is not required. |
| Subject content | Notes and examples |
|---|---|
| 1 Understand and use indices (positive, zero, negative and fractional). | e.g. solve: • $32^x = 2$ • $5^{x+1} = 25^x$. |
| 2 Understand and use the rules of indices. | e.g. simplify: • $3x^{-4} \times \frac{2}{3}x^{\frac{1}{2}}$ • $\frac{2}{5}x^{\frac{1}{2}} \div 2x^{-2}$ • $\left(\frac{2x^5}{3}\right)^3$. Knowledge of logarithms is not required. |
来源:剑桥国际大纲
指数(indices)律对字母也起作用:$a^{m} \times a^{n} = a^{m+n}$,$a^{m} \div a^{n} = a^{m-n}$,而 $(a^{m})^{n} = a^{mn}$。
Worked examples.
$$(5x^{3})^{2} = 25x^{6}, \qquad 12a^{5} \div 3a^{-2} = 4a^{7}, \qquad 6x^{7}y^{4} \times 5x^{-5}y = 30x^{2}y^{5}.$$
你也能通过把两边写成相同的底数(base)来解简单的指数方程。
Worked example. 解 $2^{x} = 32$。因为 $32 = 2^{5}$,你得到 $x = 5$。
探索
Algebraic index law lab
Classify index-law examples by the rule being used.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| indices | 指数 | zhǐ shù |
| base | 底数 | dǐ shù |
2.5
方程
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Construct simple expressions, equations and formulas. | e.g. write an expression for a number that is 2 more than $n$. Includes constructing linear simultaneous equations. |
| 2 Solve linear equations in one unknown. | Examples include: • $3x + 4 = 10$ • $5 - 2x = 3(x + 7)$. |
| 3 Solve simultaneous linear equations in two unknowns. | |
| 4 Change the subject of simple formulas. | e.g. change the subject of formulas where: • the subject only appears once • there is not a power or root of the subject. |
| Subject content | Notes and examples |
|---|---|
| 1 Construct expressions, equations and formulas. | e.g. write an expression for the product of two consecutive even numbers. Includes constructing simultaneous equations. |
| 2 Solve linear equations in one unknown. | Examples include: • $3x + 4 = 10$ • $5 - 2x = 3(x + 7)$. |
| 3 Solve fractional equations with numerical and linear algebraic denominators. | Examples include: • $\frac{x}{2x + 1} = 4$ • $\frac{2}{x + 2} + \frac{3}{2x - 1} = 1$ • $\frac{x}{x + 2} = \frac{3}{x - 6}$. |
| 4 Solve simultaneous linear equations in two unknowns. | |
| 5 Solve simultaneous equations, involving one linear and one non-linear. | With powers no higher than two. |
| 6 Solve quadratic equations by factorisation, completing the square and by use of the quadratic formula. | Includes writing a quadratic expression in completed square form. Candidates may be expected to give solutions in surd form. The quadratic formula is given in the List of formulas. |
| 7 Change the subject of formulas. | e.g. change the subject of a formula where: • the subject appears twice • there is a power or root of the subject. |
来源:剑桥国际大纲
一个方程(equation)说两个表达式相等。要解一个一次(linear)方程,对两边做相同的运算直到未知数(unknown)独立。
Worked example. 解 $5 - 2x = 3(x + 7)$。
$$5 - 2x = 3x + 21 \;\Rightarrow\; 5 - 21 = 3x + 2x \;\Rightarrow\; -16 = 5x \;\Rightarrow\; x = -\tfrac{16}{5}.$$
Fractional equations (Extended)
一个分式方程(fractional equation)在一个分母里有未知数。把两边乘以分母来清除它。
Worked example. 解 $\dfrac{x}{2x + 1} = 4$。
$$x = 4(2x + 1) = 8x + 4 \;\Rightarrow\; -7x = 4 \;\Rightarrow\; x = -\tfrac{4}{7}.$$
Simultaneous equations
联立方程(simultaneous equations)是一起求解的两个方程。对于两个一次方程,加或减以消去一个字母。
Worked example. 解 $2x + y = 7$ 和 $3x - y = 8$。
相加消去 $y$:$5x = 15$,所以 $x = 3$。然后 $y = 7 - 2(3) = 1$。
对于一个一次和一个二次方程(拓展(Extended)),把一次的代入曲线。
Worked example. 解 $y = x + 2$ 和 $y = x^{2}$。
$$x^{2} = x + 2 \;\Rightarrow\; x^{2} - x - 2 = 0 \;\Rightarrow\; (x - 2)(x + 1) = 0,$$
所以 $x = 2$(给出 $y = 4$)或 $x = -1$(给出 $y = 1$)。
Solving quadratic equations (Extended)
有三个方法。
- 通过因式分解: $x^{2} + 5x + 6 = 0 \Rightarrow (x + 2)(x + 3) = 0 \Rightarrow x = -2$ 或 $x = -3$。
- 通过配方法: $x^{2} + 6x + 1 = 0 \Rightarrow (x + 3)^{2} = 8 \Rightarrow x + 3 = \pm 2\sqrt{2} \Rightarrow x = -3 \pm 2\sqrt{2}$。
- 通过求根公式(quadratic formula),它在考试中给出:
$$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}.$$
Worked example (formula). 解 $2x^{2} + 3x - 1 = 0$。这里 $a = 2$,$b = 3$,$c = -1$:
$$x = \frac{-3 \pm \sqrt{9 + 8}}{4} = \frac{-3 \pm \sqrt{17}}{4}.$$
Changing the subject
公式变形(change the subject)一个公式的意味着重新排列它,使一个选定的字母在一边独立。
Worked example. 使 $r$ 成为 $A = \pi r^{2}$ 的主项(拓展(Extended),因为有幂)。
$$r^{2} = \frac{A}{\pi} \;\Rightarrow\; r = \sqrt{\frac{A}{\pi}}.$$
当字母出现两次时(拓展(Extended)),收集那些项并因式分解。要使 $x$ 成为 $y = \dfrac{x + 1}{x - 1}$ 的主项:
$$y(x - 1) = x + 1 \;\Rightarrow\; yx - x = 1 + y \;\Rightarrow\; x(y - 1) = 1 + y \;\Rightarrow\; x = \frac{1 + y}{y - 1}.$$
探索
Solving an equation
y = ax² + bx + c
Solving means finding the roots — where the curve crosses the x-axis.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| equation | 方程 | fāng chéng |
| linear | 一次 | yī cì |
| unknown | 未知数 | wèi zhī shù |
| fractional equation | 分式方程 | fēn shì fāng chéng |
| simultaneous equations | 联立方程 | lián lì fāng chéng |
| quadratic formula | 求根公式 | qiú gēn gōng shì |
| change the subject | 公式变形 | gōng shì biàn xíng |
2.6
不等式
大纲
| Subject content | Notes and examples |
|---|---|
| Represent and interpret inequalities, including on a number line. | When representing and interpreting inequalities on a number line: • open circles should be used to represent strict inequalities (<, >) • closed circles should be used to represent inclusive inequalities ($\leqslant$, $\geqslant$) e.g. $-3 \leqslant x < 1$ |
| Subject content | Notes and examples |
|---|---|
| 1 Represent and interpret inequalities, including on a number line. | When representing and interpreting inequalities on a number line: • open circles should be used to represent strict inequalities (<, >) • closed circles should be used to represent inclusive inequalities ($\leqslant$, $\geqslant$). e.g. $-3 \leqslant x < 1$ |
| 2 Construct, solve and interpret linear inequalities. | Examples include: • $3x < 2x + 4$ • $-3 \leqslant 3x - 2 < 7$. |
| 3 Represent and interpret linear inequalities in two variables graphically. | The following conventions should be used: • broken lines should be used to represent strict inequalities (<, >) • solid lines should be used to represent inclusive inequalities ($\leqslant$, $\geqslant$) • shading should be used to represent unwanted regions (unless otherwise directed in the question). e.g. graphs of $x < 1$ and $y \geqslant 1$ |
| 4 List inequalities that define a given region. | Linear programming problems are not included. |
来源:剑桥国际大纲
一个不等式(inequality)用 $<$、$>$、$\leqslant$ 或 $\geqslant$。像一个方程一样解它,但若你乘以或除以一个负数就反转符号。
Worked example. 解 $-3 \leqslant 3x - 2 < 7$。
对所有部分加 $2$,然后除以 $3$:
$$-1 \leqslant 3x < 9 \;\Rightarrow\; -\tfrac{1}{3} \leqslant x < 3.$$
在一条数轴(number line)上,对 $<$ 或 $>$ 用一个空心圆(值不被包含),对 $\leqslant$ 或 $\geqslant$ 用一个实心圆(值被包含)。
Regions (Extended)
一个两个字母的不等式描述图像的一个区域(region)。画边界线(对 $<$ 或 $>$ 是断的,对 $\leqslant$ 或 $\geqslant$ 是实的)并给不想要的一侧涂阴影。你也可能被要求列出定义一个给定区域的不等式。
探索
Inequalities
y = ax + b
An inequality asks where the line is above or below a value.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| inequality | 不等式 | bù děng shì |
| number line | 数轴 | shù zhóu |
| region | 区域 | qū yù |
2.7
数列
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Continue a given number sequence or pattern. | e.g. write the next two terms in this sequence: 1, 3, 6, 10, 15, ... , ... |
| 2 Recognise patterns in sequences, including the term-to-term rule, and relationships between different sequences. | |
| 3 Find and use the $n$th term of the following sequences: (a) linear (b) simple quadratic (c) simple cubic. | e.g. find the $n$th term of 2, 5, 10, 17 |
| Subject content | Notes and examples |
|---|---|
| 1 Continue a given number sequence or pattern. | Subscript notation may be used, e.g. $T_n$ is the $n$th term of sequence $T$. |
| 2 Recognise patterns in sequences, including the term-to-term rule, and relationships between different sequences. | Includes linear, quadratic, cubic and exponential sequences and simple combinations of these. |
| 3 Find and use the $n$th term of sequences. |
来源:剑桥国际大纲
一个数列(sequence)是遵循一个规则的一列数。递推规则(term-to-term rule)告诉你如何从前一项得到下一项。
要找到位置 $n$ 处的项(第 $n$ 个项(term))的规则,看这些项如何变化。
一次数列(linear sequence)(每次这些项上升相同的数额)。差是 $n$ 的倍数。
Worked example. 求 $2, 5, 8, 11, \dots$ 的第 $n$ 项。
这些项上升 $3$,所以从 $3n$ 开始。因为 $3 \times 1 = 3$ 但第一项是 $2$,减 $1$:第 $n$ 项是 $3n - 1$。
二次数列(quadratic sequence)(差本身以相同的数额变化)。二阶差等于 $2 \times$ $n^{2}$ 的系数。
Worked example. 求 $2, 5, 10, 17, \dots$ 的第 $n$ 项。
一阶差是 $3, 5, 7$;二阶差是 $2$,所以 $n^{2}$ 部分是 $1n^{2}$。从数列减去 $n^{2}$($1, 4, 9, 16$)留下 $1, 1, 1, 1$。所以第 $n$ 项是 $n^{2} + 1$。
一个像 $1, 8, 27, 64, \dots$ 这样的三次(cubic)数列有第 $n$ 项 $n^{3}$。一个像 $2, 6, 18, 54, \dots$ 这样的指数数列(exponential sequence)每次乘以一个固定的数;这里第 $n$ 项是 $2 \times 3^{\,n-1}$。
探索
Number sequences
Build an arithmetic (add d) or geometric (times r) sequence term by term.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| sequence | 数列 | shù liè |
| term-to-term rule | 递推规则 | dì tuī guī zé |
| cubic | 三次 | sān cì |
| exponential sequence | 指数数列 | zhǐ shù shù liè |
2.8
比例
大纲
| Subject content | Notes and examples |
|---|---|
| Express direct and inverse proportion in algebraic terms and use this form of expression to find unknown quantities. | Includes linear, square, square root, cube and cube root proportion. Knowledge of proportional symbol ($\propto$) is required. |
来源:剑桥国际大纲
两个量成正比例(direct proportion)若一个总是另一个的一个固定倍数:$y \propto x$ 意味着 $y = kx$,其中 $k$ 是一个常数(constant)。它们成反比例(inverse proportion)若一个随着另一个下降而上升:$y \propto \dfrac{1}{x}$ 意味着 $y = \dfrac{k}{x}$。符号 $\propto$ 读作"与……成比例"。你也能有与一个平方、平方根、立方或立方根成比例。
Worked example. $y$ 与 $x$ 成正比例,而当 $x = 3$ 时 $y = 12$。当 $x = 7$ 时求 $y$。
先求 $k$:$12 = k \times 3$,所以 $k = 4$ 而 $y = 4x$。然后 $y = 4 \times 7 = 28$。
探索
Inverse proportion
y = a/x
Inverse proportion: as x doubles, y halves — a reciprocal curve with two asymptotes.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| direct proportion | 正比例 | zhèng bǐ lì |
| constant | 常数 | cháng shù |
| inverse proportion | 反比例 | fǎn bǐ lì |
2.9
实际情境中的图像
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Use and interpret graphs in practical situations including travel graphs and conversion graphs. | e.g. interpret the gradient of a straight-line graph as a rate of change. |
| 2 Draw graphs from given data. | e.g. draw a distance–time graph to represent a journey. |
| Subject content | Notes and examples |
|---|---|
| 1 Use and interpret graphs in practical situations including travel graphs and conversion graphs. | Includes estimation and interpretation of the gradient of a tangent at a point. |
| 2 Draw graphs from given data. | |
| 3 Apply the idea of rate of change to simple kinematics involving distance–time and speed–time graphs, acceleration and deceleration. | |
| 4 Calculate distance travelled as area under a speed–time graph. | Areas will involve linear sections of the graph only. |
来源:剑桥国际大纲
一个图的斜率(gradient)(陡度)显示一个变化率(rate of change)。
- 一个行程图(travel graph)(距离-时间图)有等于速度的斜率;一个平坦的部分意味着物体不在移动。
- 一个换算图(conversion graph)是用于在两个单位之间转换的一条直线(例如,英里和千米)。
Speed–time graphs (Extended)
在一个速度-时间图上斜率是加速度(acceleration)(若速度下降则是减速度(deceleration)),而图下的面积(area)是行驶的距离(distance)。
Worked example. 一辆车在 $8\text{ s}$ 里从静止加速到 $20\text{ m/s}$,然后在 $20\text{ m/s}$ 保持 $12\text{ s}$。求加速度和总距离。
加速度 $= \dfrac{20}{8} = 2.5\text{ m/s}^{2}$。距离是面积:一个三角形加一个矩形,
$$\tfrac{1}{2} \times 8 \times 20 + 12 \times 20 = 80 + 240 = 320\text{ m}.$$
探索
Real-life graphs
y = ax + b
A distance–time or cost graph is read from its gradient and its intercept.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| gradient | 斜率 | xié lǜ |
| rate of change | 变化率 | biàn huà lǜ |
| travel graph | 行程图 | xíng chéng tú |
| conversion graph | 换算图 | huàn suàn tú |
| acceleration | 加速度 | jiā sù dù |
| deceleration | 减速度 | jiǎn sù dù |
| distance | 距离 | jù lí |
2.10
函数的图像
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Construct tables of values, and draw, recognise and interpret graphs for functions of the following forms: • $ax + b$ • $\pm x^2 + ax + b$ • $\frac{a}{x} \ (x \neq 0)$ where $a$ and $b$ are integer constants. | |
| 2 Solve associated equations graphically, including finding and interpreting roots by graphical methods. | e.g. find the intersection of a line and a curve. |
| Subject content | Notes and examples |
|---|---|
| 1 Construct tables of values, and draw, recognise and interpret graphs for functions of the following forms: • $a x^n$ (includes sums of no more than three of these) • $a b^x + c$ where $n = -2, -1, -\frac{1}{2}, 0, \frac{1}{2}, 1, 2, 3$; $a$ and $c$ are rational numbers; and $b$ is a positive integer. | Examples include: • $y = x^3 + x - 4$ • $y = 2x + \frac{3}{x^2}$ • $y = \frac{1}{4} \times 2^x$. |
| 2 Solve associated equations graphically, including finding and interpreting roots by graphical methods. | e.g. finding the intersection of a line and a curve. |
| 3 Draw and interpret graphs representing exponential growth and decay problems. |
来源:剑桥国际大纲
要画一个图,做一个数值表(table of values):选择 $x$ 的值、算出 $y$,然后描出坐标(coordinates)并用一条光滑的曲线连接它们。
图穿过 $x$ 轴(水平坐标轴(axis))的点是根(roots)——$y = 0$ 的解。
你能通过读一个图来解一个方程。一条线和一条曲线的交点(intersection point)给出两个方程一起的解。
对于指数增长(exponential growth)和指数衰减(exponential decay),$y = a\,b^{x} + c$ 的图越来越快地上升(或下降)并朝一条水平线变平。
探索
Graphing a quadratic
y = ax² + bx + c
Drag a, b and c and watch the parabola move — its turning point and where it crosses the axes.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| table of values | 数值表 | shù zhí biǎo |
| coordinates | 坐标 | zuò biāo |
| axis | 坐标轴 | zuò biāo zhóu |
| roots | 根 | gēn |
| intersection point | 交点 | jiāo diǎn |
| exponential growth | 指数增长 | zhǐ shù zēng zhǎng |
| exponential decay | 指数衰减 | zhǐ shù shuāi jiǎn |
2.11
草绘曲线
大纲
| Subject content | Notes and examples |
|---|---|
| Recognise, sketch and interpret graphs of the following functions: (a) linear (b) quadratic. | Knowledge of symmetry and roots is required. Knowledge of turning points is not required. |
| Subject content | Notes and examples |
|---|---|
| Recognise, sketch and interpret graphs of the following functions: (a) linear (b) quadratic (c) cubic (d) reciprocal (e) exponential. | Functions will be equivalent to: • $ax + by = c$ • $y = ax^2 + bx + c$ • $y = ax^3 + b$ • $y = ax^3 + bx^2 + cx$ • $y = \frac{a}{x} + b$ • $y = ar^x + b$ where $a$, $b$ and $c$ are rational numbers and $r$ is a rational, positive number. Knowledge of turning points, roots and symmetry is required. Knowledge of vertical and horizontal asymptotes is required. Finding turning points of quadratics by completing the square is required. |
来源:剑桥国际大纲
一个快速的草图应当显示正确的形状和关键特征:它在哪里穿过坐标轴、任何对称(symmetry),以及曲线接近的任何线。
| 函数 | 形状 |
|---|---|
| 一次,$y = mx + c$ | 直线;斜率 $m$、$y$-截距(intercept)$c$ |
| 二次,$y = ax^{2} + bx + c$ | 一条抛物线(parabola)(若 $a>0$ 是 U 形,若 $a<0$ 是 $\cap$ 形) |
| 三次,$y = ax^{3} + bx + c$ | 一条 S 形曲线 |
| 反比例,$y = \dfrac{a}{x} + b$ | 两条分开的曲线 |
| 指数,$y = a\,r^{x} + b$ | 快速增长或衰减 |
对于一条抛物线,配方法给出转折点(turning point)(最低或最高的点)。例如 $y = (x + 3)^{2} - 8$ 在 $(-3, -8)$ 处有它的转折点。
一条渐近线(asymptote)是曲线越来越接近但从不触及的一条线——例如,$y = \dfrac{a}{x}$ 的 $x$ 轴,或 $y = a\,r^{x} + b$ 的线 $y = b$。
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| symmetry | 对称 | duì chèn |
| intercept | 截距 | jié jù |
| parabola | 抛物线 | pāo wù xiàn |
| turning point | 转折点 | zhuǎn zhé diǎn |
| asymptote | 渐近线 | jiàn jìn xiàn |
2.12
微分
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Estimate gradients of curves by drawing tangents. | |
| 2 Use the derivatives of functions of the form $ax^n$, where $a$ is a rational constant and $n$ is a positive integer or zero, and simple sums of not more than three of these. | $\frac{\mathrm{d}y}{\mathrm{d}x}$ notation will be expected. |
| 3 Apply differentiation to gradients and stationary points (turning points). | |
| 4 Discriminate between maxima and minima by any method. | Maximum and minimum points may be identified by: • an accurate sketch • use of the second differential • inspecting the gradient either side of a turning point. Candidates are not expected to identify points of inflection. |
来源:剑桥国际大纲
微分(differentiation)求一条曲线在任何点的斜率。你能通过画一条切线(tangent)(一条恰好触及曲线的线)并测量它的斜率来估计它。
精确的规则:若 $y = ax^{n}$,那么导数(derivative)是
$$\frac{\mathrm{d}y}{\mathrm{d}x} = a\,n\,x^{\,n-1}.$$
逐项微分一个和。
Worked example. 若 $y = x^{3} + 2x^{2} - 5x$,那么 $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3x^{2} + 4x - 5$。
一个驻点(stationary point)(转折点)是斜率为零的地方,所以令 $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0$。
Worked example. 求 $y = x^{2} - 6x + 5$ 的转折点。
$$\frac{\mathrm{d}y}{\mathrm{d}x} = 2x - 6 = 0 \;\Rightarrow\; x = 3, \quad y = 3^{2} - 6(3) + 5 = -4.$$
转折点是 $(3, -4)$。要判定一个转折点是一个最大值(maximum)还是一个最小值(minimum),检查每一侧斜率的符号,或用二阶导数(正的意味着一个最小值)。
探索
Gradient of a curve
y = ax³ + bx² + cx + d
Move the point: the tangent shows the gradient there, which is what differentiation finds.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| differentiation | 微分 | wēi fēn |
| tangent | 切线 | qiè xiàn |
| derivative | 导数 | dǎo shù |
| stationary point | 驻点 | zhù diǎn |
| maximum | 最大值 | zuì dà zhí |
| minimum | 最小值 | zuì xiǎo zhí |
2.13
函数
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Understand functions, domain and range and use function notation. | Examples include: • $f(x) = 3x - 5$ • $g(x) = \frac{3(x + 4)}{5}$ • $h(x) = 2x^2 + 3$. |
| 2 Understand and find inverse functions $f^{-1}(x)$. | |
| 3 Form composite functions as defined by $gf(x) = g(f(x))$. | e.g. $f(x) = \frac{3}{x + 2}$ and $g(x) = (3x + 5)^2$. Find $fg(x)$. Give your answer as a fraction in its simplest form. Candidates are not expected to find the domains and ranges of composite functions. This topic may include mapping diagrams. |
来源:剑桥国际大纲
一个函数(function)把每个输入变成一个输出。我们写 $f(x)$,例如 $f(x) = 3x - 5$,所以 $f(2) = 1$。允许的输入的集合是定义域(domain);可能的输出的集合是值域(range)。
反函数(inverse function)$f^{-1}(x)$ 撤销函数。要找到它,写 $y = f(x)$、交换角色,并使 $x$ 成为主项。
Worked example. 求 $f(x) = 3x - 5$ 的反函数。
$$y = 3x - 5 \;\Rightarrow\; x = \frac{y + 5}{3}, \quad \text{so} \quad f^{-1}(x) = \frac{x + 5}{3}.$$
一个复合函数(composite function)一个接一个地应用一个函数:$gf(x)$ 意味着"先做 $f$,然后 $g$"。
Worked example. 若 $f(x) = 2x$ 而 $g(x) = x + 3$,那么
$$gf(x) = g(2x) = 2x + 3, \qquad fg(x) = f(x + 3) = 2(x + 3) = 2x + 6.$$
探索
Functions
y = f(x)
A function turns each input into exactly one output — watch its shape.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| function | 函数 | hán shù |
| domain | 定义域 | dìng yì yù |
| range | 值域 | zhí yù |
| inverse function | 反函数 | fǎn hán shù |
| composite function | 复合函数 | fù hé hán shù |
2.13
考试技巧
- 当你展开括号时,乘每一项并注意符号,尤其是前面有一个负号时:$-(x - 3) = -x + 3$。
- 要解一个方程,对两边做相同的事。当你把一个不等式乘以或除以一个负数时,翻转符号。
- 完全因式分解:先取出最大公因式,然后寻找一个平方差或一个二次模式。
- 一个二次式通常有两个解——给出两个。检查问题想要因式分解、公式,还是配方法。
- 当代入一个公式时,先把每个值放进括号里,这样符号和幂就出来正确。
本主题的互动课程
逐步学习,并即时检测练习。