📚 Exponential Growth and Decay | 指数增长与衰减
Exponential growth and decay are mathematical models that describe how quantities change over time at a rate proportional to their current value. These models appear everywhere in real life — from population growth and radioactive decay to bank interest and medicine dosage.
指数增长与衰减是描述一个量随时间以与其当前数值成比例的速率变化的数学模型。这些模型在现实生活中无处不在——从人口增长、放射性衰变，到银行存款利息和药物剂量。
1. What Is Exponential Growth? | 什么是指数增长？
Exponential growth occurs when a quantity increases by the same percentage or factor in each equal time period. The growth is not linear — it accelerates over time because the amount being multiplied grows larger each step.
指数增长发生在一个量在每个相等的时间段内以相同的百分比或倍数增长时。这种增长不是线性的——它会随着时间加速，因为每一步被乘上的数量本身变得越来越大。
For example, if a population of 100 bacteria doubles every hour, the numbers are 100, 200, 400, 800, and so on. Each hour the increase becomes larger, even though the doubling rate is constant at 100%.
例如，如果100个细菌每小时翻倍，那么数量就是100、200、400、800等等。尽管增长率恒定为100%，但每小时增加的绝对量却越来越大。
A key feature of exponential growth is that the ratio of consecutive values is always the same. If a quantity is growing exponentially and you divide any term by the previous term, you always get the same growth factor.
指数增长的一个关键特征是相邻数值之间的比值始终相同。如果一个量在指数增长，你拿任意一项除以前一项，总会得到相同的增长因子。
2. The Exponential Growth Formula | 指数增长公式
The standard formula for exponential growth is written as:
指数增长的标准公式写作：
N(t) = N₀ × bᵗ
Here, N(t) is the amount after time t, N₀ is the initial amount at t = 0, b is the growth factor per unit time (b > 1), and t is time.
这里，N(t)是时间t后的数量，N₀是t=0时的初始数量，b是每单位时间的增长因子（b > 1），t是时间。
When the growth is expressed as a percentage rate r, the growth factor is b = 1 + r. For example, a 5% annual growth rate means b = 1.05.
当增长用百分比r表示时，增长因子为b = 1 + r。例如，5%的年增长率意味着b = 1.05。
Sometimes the formula is written using a continuous rate k:
有时公式使用连续速率k表示：
N(t) = N₀ × eᵏᵗ
where e ≈ 2.71828 and k > 0. The two forms are equivalent when b = eᵏ. In IGCSE mathematics, the discrete form with base b is far more common, so make sure you can use it directly in calculations.
其中e ≈ 2.71828，k > 0。两种形式在b = eᵏ时等价。在IGCSE数学中，以b为底数的离散形式更为常见，所以请确保你能在计算中直接使用它。
3. The Exponential Decay Formula | 指数衰减公式
Exponential decay follows the same structure but with a decay factor between 0 and 1. The quantity decreases by a constant percentage in each time period, and the rate of decrease becomes smaller as the quantity gets smaller.
指数衰减遵循相同的结构，但衰减因子介于0和1之间。数量在每个时间段内按恒定百分比减少，并且随着数量变小，减少的速率也变得更小。
N(t) = N₀ × bᵗ, where 0 < b < 1
N(t) = N₀ × bᵗ，其中0 < b < 1
If the quantity loses 20% of its value each year, then the remaining percentage is 80%, so the decay factor is b = 1 − 0.20 = 0.80. A common mistake is to use b = 0.20; remember that b represents what remains, not what is lost.
如果一个量每年损失20%的价值，那么剩余百分数是80%，所以衰减因子是b = 1 − 0.20 = 0.80。一个常见错误是使用b = 0.20；记住b表示剩下的部分，而不是失去的部分。
For a continuous decay model, use:
对于连续衰减模型，使用：
N(t) = N₀ × e⁻ᵏᵗ
Here the negative exponent ensures that the value decreases over time. As t becomes very large, N(t) approaches 0 but never quite reaches it.
这里的负指数确保数值随时间减小。当t变得非常大时，N(t)趋近于0，但永远不会完全达到0。
4. General Form of Exponential Functions | 指数函数的一般形式
In algebra, an exponential function is often written as:
在代数中，指数函数通常写作：
y = a × bˣ
where a is the initial value when x = 0, b is the base, and x is the exponent. The base b must be positive and not equal to 1. If b > 1 the function demonstrates growth; if 0 < b < 1 it demonstrates decay.
其中a是x=0时的初始值，b是底数，x是指数。底数b必须为正且不等于1。若b > 1则函数呈增长；若0 < b < 1则函数呈衰减。
The simplest exponential function is y = 2ˣ. Its values for x = 0, 1, 2, 3 are 1, 2, 4, 8. Compare this to y = (½)ˣ whose values are 1, ½, ¼, ⅛ — a clear decay pattern.
最简单的指数函数是y = 2ˣ。它在x = 0, 1, 2, 3时的值分别是1、2、4、8。把它与y = (½)ˣ比较，其值分别是1、½、¼、⅛——一种明显的衰减模式。
Exponential functions are different from quadratic or cubic polynomial functions. A quadratic function like y = x² grows by adding larger and larger differences, but an exponential function like y = 2ˣ multiplies by the same factor each time x increases by 1.
指数函数不同于二次或三次多项式函数。像y = x²这样的二次函数通过累加越来越大的差值来增长，但像y = 2ˣ这样的指数函数每次x增加1时都乘以相同的因子。
5. Growth Factor and Decay Factor | 增长因子与衰减因子
The factor b in the formula N(t) = N₀ × bᵗ plays the central role. For growth, the factor is greater than 1. For decay, the factor is between 0 and 1. The percentage change is related to b by:
公式N(t) = N₀ × bᵗ中的因子b扮演核心角色。对于增长，该因子大于1；对于衰减，该因子介于0与1之间。百分比变化与b的关系为：
percentage change = (b − 1) × 100%
百分比变化 = (b − 1) × 100%
If b = 1.06, the quantity grows by 6% each period. If b = 0.94, the quantity decays by 6% each period. Notice that both 1.06 and 0.94 produce a 6% change, but in opposite directions.
若b = 1.06，则数量每期增长6%；若b = 0.94，则数量每期衰减6%。注意1.06和0.94都产生6%的变化，但方向相反。
When a problem says “increase by 15%”, change it to a multiplier: 1 + 0.15 = 1.15. When it says “decrease by 15%”, the multiplier is 1 − 0.15 = 0.85. Always translate percentages into multipliers before substituting into the formula.
当题目说“增加15%”时，转换成乘数：1 + 0.15 = 1.15。当题目说“减少15%”时，乘数是1 − 0.15 = 0.85。在代入公式之前，一定要把百分比转换成乘数。
| Percentage change | Multiplier b | Type |
| +10% | 1.10 | Growth |
| −30% | 0.70 | Decay |
| +100% | 2.00 | Doubling |
| −50% | 0.50 | Halving |
6. Graphs of Exponential Functions | 指数函数的图像
The graph of y = a × bˣ for b > 1 increases rapidly from left to right. It passes through the point (0, a) and rises steeply as x increases. As x approaches negative infinity, the graph approaches the x-axis but never touches it.
当b > 1时，y = a × bˣ的图像从左到右快速上升。它经过点(0, a)，随着x增大而陡峭上升。当x趋向负无穷时，图像趋近于x轴，但永远不会接触它。
For 0 < b < 1, the graph falls from left to right. It also passes through (0, a), but now as x increases, y rapidly approaches 0. The x-axis is called a horizontal asymptote because the curve gets infinitely close to it.
当0 < b < 1时，图像从左到右下降。它同样经过(0, a)，但当x增大时，y迅速趋近于0。x轴被称为水平渐近线，因为曲线无限接近它。
In an exam, you may be asked to sketch an exponential graph. Mark the y-intercept (0, a), the asymptote y = 0, and one more point such as x = 1. Make sure the curve is smooth and does not cross the x-axis.
在考试中，你可能会被要求画出指数函数的草图。标出y轴截距(0, a)、渐近线y = 0，以及另一个点比如x = 1。确保曲线平滑，并且不跨越x轴。
Graphs can also involve exponential decay that tends toward a non-zero level, such as y = 100 × 0.8ˣ + 20. Here the asymptote is y = 20, and the graph approaches this value from above.
图像也可能涉及趋向非零水平的指数衰减，例如y = 100 × 0.8ˣ + 20。这里的渐近线是y = 20，图像从上方趋近这个值。
7. Compound Interest | 复利
Compound interest is a classic application of exponential growth. When money is invested at an annual interest rate r, compounded n times per year, the balance after t years is:
复利是指数增长的一个经典应用。当资金以年利率r投资，每年复利n次时，t年后的余额为：
A = P × (1 + r/n)ⁿᵗ
where P is the principal, A is the amount after t years, r is the annual rate in decimal form, and n is the number of compounding periods per year.
其中P是本金，A是t年后的总额，r是小数形式的年利率，n是每年复利的次数。
For example, if £2000 is invested at 4% per annum compounded annually, then n = 1 and the formula simplifies to A = 2000 × 1.04ᵗ. After 3 years, A = 2000 × 1.04³ ≈ £2249.73.
例如，如果2000英镑按年利率4%每年复利一年，则n = 1，公式简化为A = 2000 × 1.04ᵗ。三年后，A = 2000 × 1.04³ ≈ 2249.73英镑。
If the interest is compounded quarterly, then n = 4 and the multiplier per period is 1 + 0.04/4 = 1.01. Over 3 years there are t × n = 12 periods, so A = 2000 × 1.01¹² ≈ £2253.65. More frequent compounding gives a slightly higher balance.
如果利息按季度复利，则n = 4，每期乘数为1 + 0.04/4 = 1.01。三年共有t × n = 12期，所以A = 2000 × 1.01¹² ≈ 2253.65英镑。更频繁的复利会产生略高的余额。
For continuous compounding, use A = P × eʳᵗ. Although IGCSE typically focuses on discrete compounding, you should recognize the continuous formula if it appears in extension work.
对于连续复利，使用A = P × eʳᵗ。虽然IGCSE通常侧重于离散复利，但如果扩展内容中出现连续公式，你应该能识别出来。
8. Radioactive Decay | 放射性衰变
Radioactive decay is a natural exponential decay process. The amount of a radioactive substance remaining after time t is modelled by:
放射性衰变是一种自然的指数衰减过程。t时间后剩余放射性物质的量可以建模为：
N(t) = N₀ × (1/2)ᵗ/H
where H is the half-life — the time needed for half the substance to decay. This formula is extremely useful because it directly uses the half-life supplied in the question.
其中H是半衰期——即一半物质衰变所需的时间。这个公式非常有用，因为它直接使用题目中给出的半衰期。
Suppose a sample starts with 80 mg of a substance with a half-life of 4 days. After 12 days, t/H = 12/4 = 3, so N = 80 × (1/2)³ = 80 × ⅛ = 10 mg.
假设一个样品最初含有80毫克这种物质，其半衰期为4天。12天后，t/H = 12/4 = 3，所以N = 80 × (1/2)³ = 80 × ⅛ = 10毫克。
You can find the half-life from raw data by solving an exponential equation. If a quantity falls from 100 to 25 in 6 years, it has halved twice in 6 years, so each half-life is 3 years.
你也可以通过求解指数方程从原始数据中找出半衰期。如果一个量在6年内从100下降到25，说明它在6年里减半了两次，所以每个半衰期是3年。
9. Population Growth | 人口增长
Population growth is often modelled with exponential functions, especially when resources are unlimited. The formula P(t) = P₀ × (1 + r)ᵗ shows how a population grows under a constant annual growth rate r.
人口增长通常使用指数函数建模，特别是在资源无限时。公式P(t) = P₀ × (1 + r)ᵗ显示了一个种群在恒定年增长率r下如何增长。
For example, a town has a population of 5000 and grows by 2% per year. After 10 years:
例如，一个小镇有5000人口，每年增长2%。十年后：
P = 5000 × 1.02¹⁰ ≈ 6094.97
So the population is approximately 6095 people. Notice that the population grew by almost 22% over 10 years, not exactly 20%, because of the compounding effect.
因此人口约为6095人。注意，十年间人口增长了大约22%，而不是精确的20%，因为存在复利效应。
To find how long a population takes to double, set P(t) = 2P₀ and solve for t. For a 2% growth rate, 2 = 1.02ᵗ, which gives t ≈ 35 years. The “rule of 70” says approximate doubling time = 70 / percentage rate, so 70/2 = 35 years.
要找到人口翻倍所需的时间，令P(t) = 2P₀并解出t。对于2%的增长率，2 = 1.02ᵗ，解得t ≈ 35年。“70法则”说明近似翻倍时间 = 70 / 百分率，所以70/2 = 35年。
10. Solving Exponential Equations | 求解指数方程
Many IGCSE questions require you to solve equations of the form a × bᵗ = c. The goal is to isolate the exponential term first, then use trial and error, logarithms, or matching bases where possible.
许多IGCSE题目要求你求解形如a × bᵗ = c的方程。目标首先是将指数部分单独分离，然后使用尝试法、对数法，或在可能时使底数相同来求解。
If b and the target quantity are powers of the same number, you can match exponents. For example, solve 2ᵗ = 32. Since 32 = 2⁵, we get t = 5 directly.
如果b和目标量是同一个数的幂，你可以匹配指数。例如，解2ᵗ = 32。因为32 = 2⁵，直接得到t = 5。
When the exponent is an unknown time and the base is not a nice integer, use logarithms. The key property is:
当指数是未知时间且底数不是整数时，使用对数。关键性质是：
if a = bᵗ, then t = log a / log b
若a = bᵗ，则t = log a / log b
For instance, solve 3ᵗ = 20. Take logs of both sides: log(3ᵗ) = log 20, so t = log 20 / log 3 ≈ 2.7268. On a calculator, use any log base, but the result is the same.
例如，解3ᵗ = 20。两边取对数：log(3ᵗ) = log 20，所以t = log 20 / log 3 ≈ 2.7268。在计算器上，使用任何对数底数，结果相同。
Be careful when the equation contains a coefficient. Solve 50 × 0.8ᵗ = 10 by first dividing both sides by 50, giving 0.8ᵗ = 0.2, then applying logs: t = log 0.2 / log 0.8 ≈ 7.21.
当方程含有系数时要小心。解50 × 0.8ᵗ = 10时，先将两边除以50，得到0.8ᵗ = 0.2，然后取对数：t = log 0.2 / log 0.8 ≈ 7.21。
11. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One common error is confusing the growth rate with the growth factor. A 25% increase means multiply by 1.25, not by 0.25. Always write percentage changes as multipliers before starting calculations.
一个常见错误是混淆增长率与增长因子。25%的增意味着乘以1.25，而不是0.25。在开始计算之前，务必把百分比变化写成乘数。
Another mistake is using the wrong number of compounding periods. In compound interest problems, the exponent is always the total number of periods (n × t), and the rate per period is r/n. Mixing these up changes the answer significantly.
另一个错误是使用错误的复利期数。在复利问题中，指数始终是总期数（n × t），每期利率是r/n。混淆这些会显著改变答案。
When drawing exponential graphs, students sometimes make the curve cross the x-axis or draw the graph as a straight line. Remember that an exponential graph never crosses its horizontal asymptote, and its slope changes continuously.
在绘制指数图像时，学生有时会让曲线穿过x轴，或把图像画成直线。记住指数图像永远不会穿过它的水平渐近线，并且其斜率在不断变化。
Finally, when solving exponential equations, do not simply divide by the base when the unknown is in the exponent. 2ᵗ = 16 cannot be solved by t = 16 ÷ 2. Instead, match bases or use logarithms.
最后，在求解指数方程时，当未知数位于指数位置时，不要简单地把两边除以底数。2ᵗ = 16不能用t = 16 ÷ 2来解。反而应该使底数相同或使用对数。
12. Summary | 总结
Exponential growth and decay are governed by the simple formula N(t) = N₀ × bᵗ. If b > 1 the quantity grows; if 0 < b < 1 it decays. The initial amount N₀ sets the starting point, and the factor b captures the percentage change per time period.
指数增长与衰减遵循简单的公式N(t) = N₀ × bᵗ。若b > 1则数量增长；若0 < b < 1则数量衰减。初始量N₀设定起点，因子b体现了每个时间段的百分比变化。
Always translate percentage rates into multipliers, choose the correct number of periods, and be aware that exponential functions approach an asymptote without reaching it. With practice, these models become straightforward to solve.
始终把百分比速率转换成乘数，选择正确的期数，并注意指数函数趋近渐近线但永远不触及它。通过练习，这些模型会变得简单易解。
Whether you are working with bank balances, populations, halflives, or cooling liquids, the exponential model gives you a powerful mathematical tool. Master it well, and you will find it repeatedly useful throughout IGCSE and beyond.
无论是研究银行余额、人口、半衰期还是液体冷却，指数模型都会为你提供强大的数学工具。学好它，你会发现它在IGCSE及其之后的学习中反复有用。
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