Exponential Functions
Introduced| English |
|---|
| base/beɪs/ |
| exponential function/ˌekspəˈnenʃl ˈfʌŋkʃn/ |
| exponential growth/ˌekspəˈnenʃl ɡrəʊθ/ |
| exponential decay/ˌekspəˈnenʃl dɪˈkeɪ/ |
| horizontal asymptote/ˌhɒrɪˈzɒntl ˈæsɪmptəʊt/ |
| initial value/ɪˈnɪʃl ˈvæljuː/ |
The base decides everything
- Put the variable up in the exponent and a whole new kind of function appears.
- $2^x$ explodes upward; $0.5^x$ melts toward zero.
- One small number — the base — flips growth into decay.
- These exponential functions model populations, money, medicine, and radioactive decay.
The shape of an exponential
- An exponential function 指数函数 is $f(x) = a\,b^x$, with base $b > 0$ and $b \neq 1$.
- The variable $x$ is the exponent; the base $b$ is fixed.
- The coefficient $a$ is the output when $x = 0$, since $b^0 = 1$.
- This is completely different from $x^2$, where the variable is the base.
An exponential function has the form…
The variable sits in the exponent over a fixed base $b$ — that is what makes it exponential, not polynomial.
Growth versus decay
- If $b > 1$, each step multiplies by more than one: exponential growth 指数增长.
- If $0 < b < 1$, each step multiplies by less than one: exponential decay 指数衰减.
- The base 底数 alone decides which way the curve goes.
- Growth curves sweep upward; decay curves fall toward the axis.
Cross the base past 1 to switch growth and decay
y = a·bˣ
Keep b above 1 for growth; drop b below 1 for decay. Notice the curve never touches the x-axis — that is its asymptote.
For $f(x) = a\,b^x$ with $a > 0$, the function shows decay when…
A base between $0$ and $1$ shrinks the output each step — exponential decay. A base above $1$ grows it.
Domain, range, and asymptote
- The domain is all real numbers — you can raise a base to any power.
- With $a > 0$ the range is $y > 0$: the output is always positive.
- The graph has a horizontal asymptote 水平渐近线 at $y = 0$, which it approaches but never touches.
- On the decay side it flattens toward the axis; on the growth side it soars.
The graph of $f(x) = a\,b^x$ has a horizontal ____ at $y = 0$ that it never crosses.
The output gets arbitrarily close to $0$ but never reaches it, giving the horizontal asymptote $y = 0$.
Select all true statements about $f(x) = a\,b^x$ (with $a>0,\ b>1$).
With $a > 0$ the output is always positive, so it never goes negative. The other three are correct.
Reading a and b in context
- The coefficient $a$ is the initial value 初始值 — the amount at time zero.
- The base $b$ is the per-step multiplier: $b = 1.08$ means $8\%$ growth each period.
- A base of $0.9$ means losing $10\%$ each period ($b < 1$, decay).
- Reading these two numbers turns the formula into a real-world story.
In $P = 500 \cdot 1.08^{\,t}$, what does the $500$ represent?
At $t = 0$, $1.08^0 = 1$, so $P = 500$: the coefficient $a$ is the initial value, and $1.08$ is the base (8% growth).
Do not confuse $a\,b^x$ (exponential) with $x^b$ (power). In $2^x$ the variable is in the exponent and growth is explosive; in $x^2$ the variable is the base and growth is merely polynomial. The exponent is what makes it exponential.
A $\$500$ deposit grows by $8\%$ a year: $P = 500 \cdot 1.08^{\,t}$.
- Initial value $a = 500$ (the balance at $t = 0$).
- Base $b = 1.08 > 1$, so this is growth of $8\%$ per year.
- After $10$ years: $P = 500 \cdot 1.08^{10} \approx \$1079$.
An exponential function $f(x) = a\,b^x$ puts the variable in the exponent. The base $b$ decides growth ($b>1$) or decay ($0