Modeling with Exponential Functions
Introduced| English |
|---|
| exponential model/ˌekspəˈnenʃl ˈmɒdl/ |
| initial value/ɪˈnɪʃl ˈvæljuː/ |
| exponential regression/ˌekspəˈnenʃl rɪˈɡreʃn/ |
| growth factor/ɡrəʊθ ˈfæktə/ |
| decay factor/dɪˈkeɪ ˈfæktə/ |
Fitting growth to real data
- A colony of bacteria, a savings account, a viral video — all grow by a constant factor.
- Given a few measurements, we want the exact rule $y = a\,b^x$ that fits them.
- Then we can predict the future and interpret the growth rate.
- This is exponential modeling: from data to a formula to a forecast.
Building the model from data
- An exponential model 指数模型 has the form $y = a\,b^x$, just like any exponential function.
- The initial value 初始值 $a$ is the amount at $x = 0$ — read it straight from the data.
- The base $b$ is the multiplier per step — find it by dividing consecutive outputs.
- Two data points are enough to pin down both $a$ and $b$.
To build $y = a\,b^x$ from data, the coefficient $a$ is the value when…
At $x = 0$, $b^0 = 1$, so $y = a$: the coefficient is the initial value.
Regression finds the best fit
- With many noisy points, use technology to run an exponential regression 指数回归.
- It finds the $a$ and $b$ whose curve comes closest to all the data at once.
- No point lands exactly on it, but the trend is captured.
- The output is a ready-to-use rule $y = a\,b^x$.
Tune the growth factor to fit the data
y = a·bˣ
Change a (the initial value) and b (the growth factor) until the curve passes through the data points.
Growth factor and decay factor
- If the base $b > 1$, it is a growth factor 增长因子: $b = 1.2$ means $+20\%$ each step.
- If $0 < b < 1$, it is a decay factor 衰减因子: $b = 0.85$ means $-15\%$ each step.
- Convert a percentage to a factor: growth $\to 1 + r$, decay $\to 1 - r$.
- Reading the factor tells you the story of the change.
If a quantity grows by $20\%$ each period, the growth ____ (the base $b$) is $1.2$.
A $20\%$ increase multiplies by $1 + 0.20 = 1.2$ — that multiplier is the growth factor.
A quantity that loses $15\%$ each period has a decay factor (base) of…
Losing $15\%$ leaves $85\%$, so you multiply by $1 - 0.15 = 0.85$ each period — the decay factor.
Predict, within limits
- Substitute an input into the model to predict an output.
- Choose a reasonable domain: negative time or absurdly large outputs are meaningless.
- Exponential growth cannot continue forever — resources, space, or money run out.
- A good model comes with a note about where it stops being trustworthy.
A colony follows $P = 100 \cdot 2^{\,t}$ (with $t$ in hours). What is $P$ at $t = 3$?
$P = 100 \cdot 2^3 = 100 \cdot 8 = 800$. Plug the input into the model to predict.
Select all true statements about exponential models.
Exponential growth extrapolated too far becomes unrealistic (nothing grows forever). The other three are correct.
Unlimited exponential growth is a fantasy. A model saying a pond's algae doubles daily will soon "predict" more algae than atoms on Earth. Every real exponential eventually slows — trust the model only over the range where its assumptions hold.
A sample of $100$ cells triples every hour: $P = 100 \cdot 3^{\,t}$.
- Initial value $a = 100$; growth factor $b = 3$ (tripling).
- After $2$ hours: $P = 100 \cdot 3^2 = 900$ cells.
- Reasonable domain: only while nutrients last — perhaps the first several hours.
An exponential model $y = a\,b^x$ has initial value $a$ and a base that is a growth factor ($b>1$) or decay factor ($0