【People's Education Press】High School Math: Compulsory Course 1 (Version A): From Isolated Points to Continuous Lines

When facing a real-world problem, we often collect discrete data. For example, the forest coverage rate of a region over the past 10 years. If we want to know what will happen in 5 or 10 years, simply staring at numbers in a table is not enough. We need a method to connect these 'isolated points' into a 'continuous line'.

This is the power ofmathematical modeling— it transforms messy data into rigorous mathematical functions through abstraction, fitting, and solving, giving us the ability to predict the future.

The Four Core Steps in Building a Function Model

In mathematical modeling, we typically follow a cyclical process aimed at finding the model that best describes real-world patterns:

Step 1: Problem Analysis and Data Collection — Identify variables and plot thescatter plotto observe distribution trends.
Step 2: Model Selection and Fitting — Choose an appropriate function prototype based on the shape of the points (straight line, parabola, exponential curve, etc.).
Step 3: Solving and Model Confirmation — Use known data points and methods like the method of undetermined coefficients to derive the analytical expression.
Step 4: Verification and Application — Return the results to the real-world context to check if they make sense or align with logic.

The modeling process is essentially a transformation from 'real-world problem → mathematical model → mathematical result → real-world conclusion.' If the model’s predictions are inaccurate, we must go back to Step 1 to re-evaluate and refine the model.

Real World $\rightleftharpoons$ Math

QUESTION 1

In the process of building a function model to solve real-world problems, what is usually the first step?

Directly perform complex formula calculations

Analyze the context, collect data, and plot a scatter plot

Write computer code

Directly write the function equation

QUESTION 2

If the points in a scatter plot roughly lie along a straight line, which model should we prioritize?

$y = ax^2 + bx + c$

$y = a \cdot b^x$

$y = kx + b$

$y = a \log_b x$

QUESTION 3

In an experiment, variable $y$ is observed to grow faster over time, showing explosive growth. Which model is most likely applicable?

Linear function model

Logarithmic function model

Exponential function model

Constant function model

QUESTION 4

When modeling, if two models have similar fit quality (error), how should we choose according to the 'principle of simplicity'?

Choose the model with a more complex equation

Choose the model with a simpler equation

Randomly select one

Do not choose any model

QUESTION 5

The primary role of a scatter plot in the modeling process is:

Directly provide prediction results

Help observe data trends to select an appropriate function type

Replace mathematical calculations

Have no practical value

QUESTION 6

After establishing a mathematical model, can the 'mathematical conclusions' be directly used as 'practical recommendations'?

Yes, mathematics is absolutely accurate

No, it must first pass a test of real-world reasonableness

Depends on mood

As long as there are many data points

QUESTION 7

Given the following data points: (1, 2.1), (2, 4.0), (3, 6.1), (4, 7.9). Which function model does this data best fit?

$y = 2x$

$y = 2^x$

$y = x^2$

$y = \log_2 x$

QUESTION 8

In the modeling process, if the independent variable $x$ represents years, the domain of $x$ in the model should typically be:

Take all real numbers $\mathbb{R}$

Determined by the actual meaningful range of years

Only take negative numbers

No need to consider the domain

QUESTION 9

If the model’s predictions deviate significantly from actual observations, you should:

Modify the observed values to fit the model

Abandon modeling

Re-examine the data and try switching to a more suitable function model

Forcefully explain the deviation as normal

Year $x$	1	2	3	4	5
Population $y$	10.2	20.1	39.8	80.3	160.5