Polynomial Modeling

Polynomial modeling is the process of finding a polynomial function that closely fits a set of real-world data. By creating these mathematical models, we can identify trends, make predictions, and optimize outcomes.

Selecting the Appropriate Degree

The first step in modeling is determining the degree of the polynomial (linear, quadratic, cubic, etc.). For evenly spaced $x$-values, you can use the method of finite differences:

• If the first differences in $y$-values are constant, the data is linear (degree 1).
• If the second differences are constant, the data is quadratic (degree 2).
• If the third differences are constant, the data is cubic (degree 3).

Example: Finding a Model from Data

Suppose a dataset shows the following values:

• $x = 0, 1, 2, 3, 4$
• $y = 2, 5, 14, 29, 50$

Let's find the differences between consecutive $y$-values:

1. First differences: $5-2 = 3$, $14-5 = 9$, $29-14 = 15$, $50-29 = 21$
2. Second differences: $9-3 = 6$, $15-9 = 6$, $21-15 = 6$

Since the second differences are constant, a quadratic model ($y = ax^2 + bx + c$) is perfect for this data.

To find the exact equation, use the data points:

• At $x = 0$, $y = 2$, so $c = 2$.
• At $x = 1$, $y = 5$, so $a(1)^2 + b(1) + 2 = 5 \implies a + b = 3$.
• At $x = 2$, $y = 14$, so $a(2)^2 + b(2) + 2 = 14 \implies 4a + 2b = 12 \implies 2a + b = 6$.

Subtracting the two equations gives $a = 3$. Substituting $a = 3$ back in gives $b = 0$.

The polynomial model is: $y = 3x^2 + 2$

Regression and Technology

Real-world data is rarely perfect. When finite differences aren't exactly constant, we use polynomial regression via graphing calculators or software. These tools use a method called "least squares" to find the curve of best fit.

When using technology, visually inspect a scatter plot of your data. If the data has one turning point (a single peak or valley), a quadratic model (degree 2) is usually best. If it has two turning points, try a cubic model (degree 3).

Interpreting the Model and Domain

A mathematical model is only useful within a logical context. The valid domain restricts the $x$-values to those that make sense in the real world (e.g., time and physical quantities cannot usually be negative).

Example: Optimizing Revenue

Consider a factory where the revenue based on production level $x$ (in thousands of units) is modeled by: $R(x) = -2x^3 + 15x^2 + 12x$

To find the production level that maximizes revenue, we need to find the local maximum of this cubic function. In Grade 11, you can find this by graphing the function and locating the peak (the turning point).

Using a graphing tool, the highest point on the graph occurs at approximately $x \approx 5.37$.

Interpretation:

• The factory should produce about $5,370$ units to maximize revenue.
• The valid domain for this model is $x \ge 0$, because producing negative units is impossible.
• Extrapolation warning: If we look too far to the right on the graph, $R(x)$ eventually becomes negative. The model is only valid up to the point where revenue drops to zero.