10-8 Modeling and Curve Fitting

Property

When analyzing a table of data where the $x$-values are evenly spaced (e.g., increasing by 1), you can identify the underlying function family by analyzing the patterns in the $y$-values:

• Linear Model: The first differences (the difference between consecutive $y$-values) are constant. This means the data grows by adding the same amount each time.
• Quadratic Model: The first differences are not constant, but the second differences (the difference between consecutive first differences) are constant and non-zero.
• Exponential Model: The differences are not constant, but the successive ratios (dividing a $y$-value by the previous $y$-value) are constant. This means the data grows by multiplying by the same factor each time.

Examples

• Linear (Constant First Difference): Data $(1, 3), (2, 7), (3, 11), (4, 15)$.

First differences: $7-3 = 4$; $11-7 = 4$; $15-11 = 4$. The constant difference is 4.

• Quadratic (Constant Second Difference): Data $(0, 1), (1, 4), (2, 9), (3, 16)$.

First differences: $3, 5, 7$.
Second differences: $5-3 = 2$; $7-5 = 2$. The constant second difference is 2.

• Exponential (Constant Ratio): Data $(0, 3), (1, 6), (2, 12), (3, 24)$.

First differences are $3, 6, 12$ (not constant).
Successive ratios: $6/3 = 2$; $12/6 = 2$; $24/12 = 2$. The constant ratio is 2.

Explanation

Think of this process as running a diagnostic test on mysterious data. By systematically checking how the $y$-values change, you reveal the data's "genetic code." Addition patterns point to lines, multiplying patterns point to exponential curves, and a constant rate of change in the rate of change points to a parabola. Always check these patterns in order: first differences, then second differences, then ratios.

Property

Once you identify the pattern in a perfect data table (where $x$ starts at 0 and increases by 1), you can manually calculate the parameters of the function:

• Linear ($y = mx + b$): $m$ is the constant first difference. $b$ is the $y$-value when $x = 0$.
• Quadratic ($y = ax^2 + bx + c$): $a$ is EXACTLY HALF of the constant second difference ($a = \frac{\Delta^2}{2}$). $c$ is the $y$-value when $x = 0$. Substitute a known point to find $b$.
• Exponential ($y = ab^x$): $b$ is the constant successive ratio. $a$ is the $y$-value when $x = 0$.

Examples

• Linear: Table has $x = 0, 1, 2$ and $y = 5, 8, 11$.

Constant first difference is 3, so $m = 3$. At $x=0$, $y=5$, so $b=5$. Model: $y = 3x + 5$.

• Quadratic: Table has $x = 0, 1, 2, 3$ and $y = 3, 6, 13, 24$.

First differences: $3, 7, 11$. Second differences: $4, 4$.
So, $a = 4 / 2 = 2$. At $x=0$, $y=3$, so $c=3$.
Substitute point $(1, 6)$ into $y = 2x^2 + bx + 3 \rightarrow 6 = 2(1)^2 + b(1) + 3 \rightarrow b = 1$. Model: $y = 2x^2 + x + 3$.

• Exponential: Table has $x = 0, 1, 2$ and $y = 4, 12, 36$.

Constant ratio is 3, so base $b = 3$. At $x=0$, $y=4$, so initial value $a = 4$. Model: $y = 4(3)^x$.

Explanation

When real-world data is mathematically perfect, you don't need a computer to find the equation. The $y$-intercept (where $x=0$) always gives you your starting parameter ($b$ for linear, $c$ for quadratic, $a$ for exponential). The constant pattern you found in your diagnostic test directly provides your growth parameter. For quadratic models, just remember the golden rule: the leading coefficient '$a$' is always half of the second difference!

Property

When modeling real-world data recorded in calendar years (e.g., 2015, 2016, 2017), you must define a shifted input variable $t$ so that the modeling begins at $t = 0$:

The baseline year is typically the first year in your data set. Use $t$ as your input variable for all calculations and regressions.