Choosing a Function Model
Introduced| English | Meaning |
|---|---|
| linear model/ˈlɪnɪə ˈmɒdl/ | A model that describes a relationship using a linear equation. |
| quadratic model/kwɒˈdrætɪk ˈmɒdl/ | A model that describes a relationship using a polynomial of degree two. |
| rate of change/reɪt ɒv tʃeɪndʒ/ | The change in one quantity divided by the corresponding change in another. |
| concavity/kənˈkævɪti/ | The direction in which a curve bends, indicated by how its gradient changes. |
| end behavior/end bɪˈheɪvjə/ | The way a function behaves as its input becomes very large positively or negatively. |
| assumptions/əˈsʌmpʃnz/ | Statements accepted as starting points for reasoning, analysis or a model. |
Which curve should we trust?
- Real data never lands perfectly on a line or curve — it scatters.
- Our job is to pick the family of function that best captures its trend.
- Choose a line, a parabola, or an exponential — each tells a different story.
- The data itself drops clues about which one to reach for.
Match the shape to the data
- First, plot the data and look at its overall shape.
- Rising by a steady amount → a linear model 线性模型.
- Curving with a bend → a quadratic model 二次模型 or another curved family.
- Levelling off toward a ceiling → something that flattens, not a line.
Data that rises by roughly the same amount each step is best fit by a…
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A constant rate of change is the signature of a linear model — a straight line.
Read the clues
- The rate of change 变化率: constant → linear; changing → curved.
- The concavity 凹凸性: a bend upward or downward rules a straight line out.
- The end behavior 末端行为: does the trend keep growing, or settle toward a limit?
- Together these three features narrow the choice fast.
How well does a straight line fit?
Strong, straight-line data has a correlation near 1 — a sign that a linear model is a good choice.
Data that curves upward, rising faster and faster, suggests a…
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An increasing rate of change (concave up) rules out a line — a quadratic or exponential curve fits better.
Which features help you choose a model?
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Rate of change, concavity, and end behavior all point to a model family; the point colour is irrelevant.
State the assumptions
- Every model makes simplifying assumptions 假设 — say them out loud.
- "Growth stays steady", "no sudden shocks", "the pattern continues" are all assumptions.
- Choose a sensible domain: a population model should not allow negative time.
- Naming the limits tells a reader exactly when the model can be trusted.
A responsible model states the ____ and restrictions it places on the situation.
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Every model simplifies reality; naming the assumptions tells the reader when to trust it.
Every model has limits
- A model is a useful approximation, never the whole truth.
- It fits well within the data but can drift badly far outside it.
- Two different families may fit the same points almost equally — extra knowledge breaks the tie.
- Always ask: does this prediction still make sense in the real situation?
A model is an exact description of reality that is always correct.
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A model is a useful approximation with limits — it can fail outside the data or when its assumptions break.
Do not pick a model just because it passes close to the points. A high-degree polynomial can thread through every data point yet swing wildly between them. Prefer the simplest family whose shape genuinely matches the trend.
A shop's daily sales for a week: $12, 15, 18, 21, 24, 27, 30$.
- Each day rises by exactly $3$ — a constant rate of change.
- Constant rate → a linear model: $\text{sales} = 12 + 3d$.
- Assumption: the steady $+3$ trend continues; realistic only for a short while.
Choose a model by matching its shape to the data, using the rate of change, concavity, and end behavior as clues: steady rate → linear model, a bend → quadratic model, and so on. State your assumptions and a sensible domain, and remember every model has limits.
Write from a linear model
- Successive outputs rise by 4, then 4, then 4.
- Name the model that matches a constant rate.
A data set rises by 4, then by 4 again, then by 4 again. Which model shape matches that rate, linear, quadratic or exponential?
Equal steps of 4 are a constant rate.
Which model fits equal steps of 4?
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A constant rate is linear.