Determine whether the data has the add-add, add-multiply,...

Determine whether the data has the add-add, add-multiply, multiply-multiply, or constant-second-differences pattern. Identify the type of function that has the pattern.
$$\begin{array}{rr}
\frac{x}{2} & f(x) \\
4 & 2700 \\
6 & 2400 \\
8 & 2100 \\
10 & 1800 \\
1500
\end{array}$$

Key Concepts

Difference Analysis

This concept involves examining the differences between consecutive output values of a function to identify patterns. It is a vital tool in determining the type of function represented by a sequence of data. For example, constant first differences indicate a linear relationship, while constant second differences are characteristic of a quadratic function. More complex patterns, such as combinations of additive and multiplicative changes, can suggest other non-linear relationships.

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. When a function's outputs form an arithmetic sequence, it means that the function changes by the same amount for equal increments in the input. This concept relates directly to an 'add-add' pattern where the operation applied between terms is repeated addition (or subtraction), typically indicating a linear function.

Linear Functions

Linear functions are defined by a constant rate of change, meaning that the difference between any two consecutive outputs is the same over uniform increases in the input. Recognizing a linear function by examining the differences in output values (which remain constant) is a fundamental approach in algebra and is closely linked to the concept of arithmetic sequences.

Constant Second Differences

When the differences of the first differences (known as second differences) are constant, it is an indication that the function is quadratic. This pattern emerges because quadratic functions have a changing rate of first differences that themselves increase or decrease linearly, resulting in a uniform second difference, which is a hallmark of second-degree polynomial functions.

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