代数与图像

In algebra 代数 we use letters to stand for numbers. A letter whose value can change is a variable 变量. To substitute 代入 means to put a number in place of a letter.

Worked example. Find the value of $3x^{2} - 2y$ when $x = 4$ and $y = 5$.

A term 项 is a single part of an expression 表达式, such as $5a$ or $-9b$. Like terms 同类项 have the same letters raised to the same powers; you may add or subtract them. The number in front of the letter is the coefficient 系数.

Worked example. Simplify $2a^{2} + 3ab - 1 + 5a^{2} - 9ab + 4$.

Collect like terms: $2a^{2} + 5a^{2} = 7a^{2}$, $3ab - 9ab = -6ab$, $-1 + 4 = 3$. So the answer is

To expand 展开 means to multiply out brackets 括号. Multiply every term inside by the term outside; for two brackets, multiply every term in the first by every term in the second.

Worked examples.

Common mistakes. Multiplying only the first term inside a bracket ($3(x + 4) = 3x + 12$, not $3x + 4$), losing a minus sign ($-2(x - 5) = -2x + 10$), and writing $(x + 3)^{2} = x^{2} + 9$. Squaring a bracket gives three terms: $(x + 3)^{2} = x^{2} + 6x + 9$.

For three brackets (Extended), expand two first, then multiply by the third:

To factorise 因式分解 is the opposite of expanding: write the expression as a product of brackets. Always take out the common factor 公因式 first.

Worked example. $9x^{2} + 15xy = 3x(3x + 5y)$, because $3x$ divides both terms.

The following patterns are Extended.

Grouping (four terms): take a common factor from each pair.

Difference of two squares 平方差: $a^{2} - b^{2} = (a + b)(a - b)$.

Perfect square 完全平方: $a^{2} + 2ab + b^{2} = (a + b)^{2}$.

Quadratic 二次 expressions $ax^{2} + bx + c$: find two numbers that multiply to $a \times c$ and add to $b$, then split the middle term.

Worked example. Factorise $2x^{2} + 7x + 3$.

Here $a \times c = 6$ and $b = 7$. The numbers $1$ and $6$ work. Split and group:

For $ax^{3} + bx^{2} + cx$, take out the common $x$ first: $2x^{3} + 7x^{2} + 3x = x(2x^{2} + 7x + 3) = x(x + 3)(2x + 1)$.

Completing the square 配方法 rewrites $x^{2} + bx + c$ as $(x + p)^{2} + q$. Take half of the $x$-coefficient, square it, then balance.

Worked example. Write $x^{2} + 6x + 1$ in completed-square form.

Half of $6$ is $3$, and $3^{2} = 9$:

When the $x^{2}$ has a coefficient, take it out of the first two terms first:

An algebraic fraction 分式 has algebra on the top or bottom. Add and subtract using a common denominator; multiply and divide as with ordinary fractions.

Worked examples.

To simplify a rational expression 有理式, factorise the top and bottom, then cancel common brackets.

The laws of indices 指数 work with letters too: $a^{m} \times a^{n} = a^{m+n}$, $a^{m} \div a^{n} = a^{m-n}$, and $(a^{m})^{n} = a^{mn}$.

Worked examples.

You can also solve simple index equations by writing both sides with the same base 底数.

Worked example. Solve $2^{x} = 32$. Since $32 = 2^{5}$, you get $x = 5$.

An equation 方程 says two expressions are equal. To solve a linear 一次 equation, do the same operation to both sides until the unknown 未知数 is alone.

Worked example. Solve $5 - 2x = 3(x + 7)$.

Common mistake. Moving a term across the equals sign without changing its sign. Do the same thing to both sides instead: "subtract $21$ from both sides" makes the sign change automatic.

Fractional equations (Extended)

A fractional equation 分式方程 has the unknown in a denominator. Multiply both sides by the denominator to clear it.

Worked example. Solve $\dfrac{x}{2x + 1} = 4$.

Simultaneous equations

Simultaneous equations 联立方程 are two equations solved together. For two linear equations, add or subtract to remove one letter.

Worked example. Solve $2x + y = 7$ and $3x - y = 8$.

Adding removes $y$: $5x = 15$, so $x = 3$. Then $y = 7 - 2(3) = 1$.

For one linear and one quadratic equation (Extended), substitute the linear into the curve.

Worked example. Solve $y = x + 2$ and $y = x^{2}$.

so $x = 2$ (giving $y = 4$) or $x = -1$ (giving $y = 1$).

Solving quadratic equations (Extended)

There are three methods.

• By factorising: $x^{2} + 5x + 6 = 0 \Rightarrow (x + 2)(x + 3) = 0 \Rightarrow x = -2$ or $x = -3$.
• By completing the square: $x^{2} + 6x + 1 = 0 \Rightarrow (x + 3)^{2} = 8 \Rightarrow x + 3 = \pm 2\sqrt{2} \Rightarrow x = -3 \pm 2\sqrt{2}$.
• By the quadratic formula 求根公式, which is given in the exam:
  $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}.$$

Worked example (formula). Solve $2x^{2} + 3x - 1 = 0$. Here $a = 2$, $b = 3$, $c = -1$:

Changing the subject

To change the subject 公式变形 of a formula means to rearrange it so a chosen letter is alone on one side.

Worked example. Make $r$ the subject of $A = \pi r^{2}$ (Extended, because of the power).

When the letter appears twice (Extended), collect those terms and factorise. To make $x$ the subject of $y = \dfrac{x + 1}{x - 1}$:

An inequality 不等式 uses $<$, $>$, $\leqslant$ or $\geqslant$. Solve it like an equation, but reverse the sign if you multiply or divide by a negative number.

Worked example. Solve $-3 \leqslant 3x - 2 < 7$.

Add $2$ to all parts, then divide by $3$:

Worked example. Solve $5 - 2x > 11$.

Dividing by $-2$ reverses the sign. Check with $x = -4$: $5 + 8 = 13 > 11$, which is correct.

Common mistake. Forgetting to reverse the sign when dividing by a negative number, which here would give the wrong answer $x > -3$.

On a number line 数轴, use an open circle for $<$ or $>$ (value not included) and a closed circle for $\leqslant$ or $\geqslant$ (value included).

Regions (Extended)

An inequality in two letters describes a region 区域 of the graph. Draw the boundary line (broken for $<$ or $>$, solid for $\leqslant$ or $\geqslant$) and shade the unwanted side. You may also be asked to list the inequalities that define a given region.

A sequence 数列 is a list of numbers that follow a rule. The term-to-term rule 递推规则 tells you how to get the next term from the one before.

To find the rule for the term in position $n$ (the $n$th term), look at how the terms change.

Linear sequence (the terms go up by the same amount each time). The difference is the multiple of $n$.

Worked example. Find the $n$th term of $2, 5, 8, 11, \dots$

The terms go up by $3$, so start with $3n$. Since $3 \times 1 = 3$ but the first term is $2$, subtract $1$: the $n$th term is $3n - 1$.

Quadratic sequence (the differences themselves change by the same amount). The second difference equals $2 \times$ the coefficient of $n^{2}$.

Worked example. Find the $n$th term of $2, 5, 10, 17, \dots$

First differences are $3, 5, 7$; the second difference is $2$, so the $n^{2}$ part is $1n^{2}$. Subtracting $n^{2}$ ($1, 4, 9, 16$) from the sequence leaves $1, 1, 1, 1$. So the $n$th term is $n^{2} + 1$.

A cubic 三次 sequence such as $1, 8, 27, 64, \dots$ has $n$th term $n^{3}$. An exponential sequence 指数数列 such as $2, 6, 18, 54, \dots$ multiplies by a fixed number each time; here the $n$th term is $2 \times 3^{\,n-1}$.

Two quantities are in direct proportion 正比例 if one is always a fixed multiple of the other: $y \propto x$ means $y = kx$, where $k$ is a constant 常数. They are in inverse proportion 反比例 if their product is constant: $y \propto \dfrac{1}{x}$ means $y = \dfrac{k}{x}$. The symbol $\propto$ is read "is proportional to". You can also have proportion to a square, square root, cube or cube root.

Worked example. $y$ is in direct proportion to $x$, and $y = 12$ when $x = 3$. Find $y$ when $x = 7$.

First find $k$: $12 = k \times 3$, so $k = 4$ and $y = 4x$. Then $y = 4 \times 7 = 28$.

The gradient 斜率 (steepness) of a graph shows a rate of change 变化率.

• A travel graph 行程图 (distance–time graph) has gradient equal to speed; a flat part means the object is not moving.
• A conversion graph 换算图 is a straight line used to change between two units (for example, miles and kilometres).

Speed–time graphs (Extended)

On a speed–time graph the gradient is the acceleration 加速度 (or deceleration 减速度 if the speed falls), and the area 面积 under the graph is the distance 距离 travelled.

Worked example. A car speeds up from rest to $20\text{ m/s}$ in $8\text{ s}$, then stays at $20\text{ m/s}$ for $12\text{ s}$. Find the acceleration and the total distance.

Acceleration $= \dfrac{20}{8} = 2.5\text{ m/s}^{2}$. The distance is the area: a triangle plus a rectangle,

To draw a graph, make a table of values 数值表: choose values of $x$, work out $y$, then plot the coordinates 坐标 and join them with a smooth curve.

The points where a graph crosses the $x$-axis (the horizontal axis 坐标轴) are the roots 根 — the solutions of $y = 0$.

You can solve an equation by reading a graph. The intersection point 交点 of a line and a curve gives the solution of the two equations together.

For exponential growth 指数增长 and exponential decay 指数衰减, consider $y = a\,b^{x} + c$ with $a>0$ and $b>0$, $b\ne1$. If $b>1$, the graph grows as $x$ increases and approaches $y=c$ as $x$ decreases. If $0<b<1$, it decays towards $y=c$ as $x$ increases.

A quick sketch should show the right shape and the key features: where it crosses the axes, any symmetry 对称, and any line the curve gets close to.

For a parabola, completing the square gives the turning point 转折点 (the lowest or highest point). For example $y = (x + 3)^{2} - 8$ has its turning point at $(-3, -8)$.

An asymptote 渐近线 is a line that the curve approaches in a stated limit. A curve may cross a horizontal asymptote elsewhere. In these examples the curve does not meet the stated line — for example, the $x$-axis for $y = \dfrac{a}{x}$, or the line $y = b$ for $y = a\,r^{x} + b$.

Differentiation 微分 finds the gradient of a curve where that gradient exists. You can estimate it by drawing a tangent 切线 (a line that just touches the curve) and measuring its gradient.

The exact rule: if $y = ax^{n}$, then the derivative 导数 is

Differentiate a sum term by term.

Worked example. If $y = x^{3} + 2x^{2} - 5x$, then $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3x^{2} + 4x - 5$.

A stationary point 驻点 is where the gradient is zero. It need not be a turning point: $y=x^3$ keeps increasing through its stationary point at the origin. To find stationary points, set $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0$.

Worked example. Find the turning point of $y = x^{2} - 6x + 5$.

The turning point is $(3, -4)$. To decide whether a turning point is a maximum 最大值 or a minimum 最小值, check the sign of the gradient on each side, or use the second derivative (positive means a minimum).

A function 函数 turns each input into one output. We write $f(x)$, for example $f(x) = 3x - 5$, so $f(2) = 1$. The set of allowed inputs is the domain 定义域; the set of possible outputs is the range 值域.

The inverse function 反函数 $f^{-1}(x)$ undoes the function. Each output must come from only one allowed input; restrict the original domain if necessary. To find it, write $y = f(x)$, swap the roles, and make $x$ the subject.

Worked example. Find the inverse of $f(x) = 3x - 5$.

A composite function 复合函数 applies one function after another: $gf(x)$ means "do $f$ first, then $g$".

Worked example. If $f(x) = 2x$ and $g(x) = x + 3$, then

Common mistakes. Doing the outside function first ($gf(x)$ starts with $f$, the one nearest to $x$), and confusing $f^{-1}(x)$ with $\dfrac{1}{f(x)}$. For $f(x) = 3x - 5$, $f^{-1}(1) = 2$ but $\dfrac{1}{f(1)} = -\dfrac{1}{2}$.

• When you expand brackets, multiply every term and watch the signs, especially with a minus in front: $-(x - 3) = -x + 3$.
• To solve an equation, do the same thing to both sides. When you multiply or divide an inequality by a negative number, flip the sign.
• Factorise fully: take out the highest common factor first, then look for a difference of two squares or a quadratic pattern.
• A quadratic usually has two solutions — give both. Check whether the question wants factorising, the formula, or completing the square.
• When substituting into a formula, put each value in brackets first, so signs and powers come out right.