Building and Applying a Model
Introduced| English |
|---|
| function model/ˈfʌŋkʃn ˈmɒdl/ |
| regression/rɪˈɡreʃn/ |
| interpolation/ɪnˌtɜːpəˈleɪʃn/ |
| extrapolation/ekˈstræpəleɪʃn/ |
| parameters/pəˈræmɪtəz/ |
From data to a formula
- Selecting a model family is only half the job; now you build the actual rule.
- A good model turns a table of measurements into a formula you can compute with.
- Once you have the formula, you can predict values you never measured.
- This is how precalculus becomes a tool for the real world.
Build the model
- A function model 函数模型 is a rule that captures how one quantity depends on another.
- You can build it from a verbal description, a table of values, or a graph.
- "Costs $\$5$ to start plus $\$2$ per item" becomes $C = 5 + 2n$ straight away.
- Match the family to the shape (Lesson 1.13), then pin down the exact numbers.
Regression finds the best fit
- With many scattered points, regression 回归 finds the best-fit line or curve using technology.
- It chooses the model that comes closest to all the data at once.
- The result gives you the numbers in the model rule, ready to use.
- No single point sits on it perfectly — it captures the overall trend.
A model line — read its rate and starting value
y = ax + b
Here a is the rate of change and b is the starting value. Adjust them to fit data, then plug in an input to make a prediction.
A regression finds…
Regression fits the model that comes closest to all the points at once — the best-fit model, using technology.
Interpolate versus extrapolate
- Interpolation 内插 predicts between known data points — usually safe.
- Extrapolation 外推 predicts beyond the data — riskier the further you go.
- A model with no data to lean on can drift far from reality.
- Trust interpolation more than extrapolation, and say which one you used.
Predicting a value between known data points is called ____.
Interpolation stays inside the data range, so it is usually reliable.
Predicting far outside the range of the data (extrapolation) is usually…
Far from the data the model has no evidence to lean on, so extrapolation is the riskiest prediction.
Interpret the parameters
- The parameters 参数 (the numbers in the rule) carry real-world meaning.
- In $C = 5 + 2n$, the $5$ is the fixed start-up cost and the $2$ is the cost per item.
- Reading parameters in context turns a formula back into a story about the situation.
- A good answer always says what each number means, with its units.
A model gives cost $= 2x + 1$ (in thousands). Predict the cost when $x = 10$.
Substitute: $2(10) + 1 = 21$ thousand. Plugging an input into the model is how you apply it.
The parameters of a model (like the slope) can be interpreted in terms of the real situation.
In cost $= 2x + 1$, the $2$ is the cost per unit and the $1$ is the fixed start-up cost — real meanings.
Put the steps of using a model in order.
Data → fit → predict → check. The final check catches a model that has drifted from reality.
Extrapolating too far is a classic trap. A linear sales model may predict billions of sales in twenty years — clearly nonsense. Always sanity-check a prediction against the real situation before trusting it.
A phone battery drops from $100\%$ by about $8\%$ per hour.
- Model: $B = 100 - 8h$, a linear function model.
- Interpolate at $h = 3$: $B = 100 - 24 = 76\%$ — inside the data, reliable.
- Extrapolate at $h = 20$: $B = -60\%$ — impossible, so the model has broken down.
Build a function model from words, a table, or a graph, using regression for the best fit. Use it to interpolate (safe, inside the data) or extrapolate (risky, beyond it), interpret its parameters in context, and always check the prediction makes real-world sense.