Recall that for linear equations, first differences are...

Recall that for linear equations, first differences are constant; and that for quadratic equations, second differences are constant. Determine whether the relationship in each table could be linear, quadratic, or neither. $$ \begin{array}{|c|c|}\hline x & {y} \\ \hline-3 & {0} \\ {-2} & {-0.5} \\ {-1} & {-1} \\ {0} & {-1.5} \\ {1} & {-2} \\ {2} & {-2.5} \\ \hline\end{array} $$

Recall that for linear equations, first differences are constant; and that for quadratic equations, second differences are constant. Determine whether the relationship in each table could be linear, quadratic, or neither.
$$
\begin{array}{|c|c|}\hline x & {y} \\ \hline-3 & {0} \\ {-2} & {-0.5} \\ {-1} & {-1} \\ {0} & {-1.5} \\ {1} & {-2} \\ {2} & {-2.5} \\ \hline\end{array}
$$

Key Concepts

Difference Tables

Difference tables are a method for analyzing sequences of numerical data by computing the differences between successive values. The first row of differences helps identify linear behavior, while subsequent rows, especially the second row, are useful for detecting quadratic patterns. This technique simplifies the process of classifying the algebraic nature of the relationship based on observed consistency in these differences.

Quadratic Functions and Constant Second Differences

A quadratic function is described by an expression in which the highest power of the variable is two. For such functions, while the first differences between consecutive y-values may vary, the differences of these differences (known as second differences) are constant when x-values are equally spaced. This characteristic is crucial when analyzing a table of values to decide if the underlying relationship is quadratic.

Linear Functions and Constant First Differences

A linear function is one in which the relationship between the independent variable (x) and the dependent variable (y) is characterized by a constant rate of change. In tabular data, this means that the differences between consecutive y-values (first differences) remain constant when the x-values are equally spaced. This property is used to determine if a set of data points might represent a linear relationship.

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