Lesson 5: Comparing Linear, Exponential, and Quadratic Models

Property

Before analyzing differences or ratios in data, verify that consecutive x-values have constant differences: $\Delta x = x_2 - x_1 = x_3 - x_2 = x_4 - x_3 = \ldots$

Examples

Property

When data follows a linear pattern, the first differences between consecutive $y$-values are constant.

For data points with evenly spaced $x$-values, if $\Delta y = y_{i+1} - y_i$ is the same for all consecutive pairs, then the data can be modeled by a linear function.

Property

Second differences are calculated by finding the differences between consecutive first differences: $\text{Second difference} = (\text{first difference}_2) - (\text{first difference}_1)$.
When second differences are constant and non-zero, the data follows a quadratic pattern.

Examples

Property

Exponential functions have constant ratios between consecutive y-values when x-values have constant differences.
For exponential data, $\frac{y_2}{y_1} = \frac{y_3}{y_2} = \frac{y_4}{y_3} = r$ (constant ratio), and the general form is $y = ab^x$ where $a$ is the initial value and $b$ is the growth/decay factor.

Examples