Lesson 6: Comparing Linear, Exponential, and Quadratic Functions

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

The average rate of change of a function between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

Examples